Section 1 (1.1 - 1.3)

1.3 Evaluate the following expression by drawing the unit circle and the appropriate right triangle. sinπ/6 Part 1 sinπ6=_____ ​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

1/2

1.1 If ​f(x)=1x2+2​, what is ​f(3​)? What is fm2​? ​f(3​)=____ f(m^2)=____​ ​(Type an integer or a simplified​ fraction.)

​f(3​)=3​)= 1/11 ​ f(m^2)= 1/(m^4)+2

1.2 What is the domain of a rational​ function? Question content area bottom Part 1 Choose the correct answer below. A. The domain of a rational function is the set of all real numbers except zero. B. The domain of a rational function is the set of all real numbers except those for which the denominator is zero. C. The domain of a rational function is the set of all rational numbers except zero. D. The domain of a rational function is the set of all real numbers that are rational numbers.

B. The domain of a rational function is the set of all real numbers except those for which the denominator is zero.

1.3 Where is the tangent function​ undefined? Question content area bottom Part 1 Choose the correct answer below. A. {x: x is an odd multiple of π} B. {x: x is an even multiple of π} C. x: x is an odd multiple of π2 D. x: x is an even multiple of π2

C. x: x is an odd multiple of π/2 Note: The tangent function is undefined for any odd multiple of π/2. Odd multiples of π/2 have a cosine of​ 0, which makes the tangent function undefined.

1.3 Find the exact value of each of the remaining trigonometric functions of θ. cosθ=−45​, θ in quadrant II

part 1 sinθ=three fifths 3/5 Part 2 tanθ=negative three fourths −3/4 Part 3 cotθ=negative four thirds −4/3 Part 4 secθ=negative five fourths −5/4 Part 5 cscθ=five thirds 5/3

1.2 example Determine the function f represented by the graph of the line y=​f(x) in the figure. 0-10-5510-10-5510xy(0,−1)(8,−8) A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A line falls from left to right, passing through the plotted points (0, negative 1) and (8, negative 8). Question content area bottom Part 1 A

A linear function has the form ​f(x)=mx+b. Use the two points given in the graph to find the slope of the linear function. m = −8−(−1)8−0 = −78 Part 2 The​ y-intercept of the function is ​(0,−1​) so b=−1. Part 3 Substitute m and b into the linear function. ​f(x) = mx+b = −78x+​(−1​) Part 4 The linear function that corresponds to the given graph is ​f(x)=−78x−1.

1.3 Use an identity to solve the equation on the interval [0,2π). sinxcosx=1/4 Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. x=enter your response here ​(Type an exact answer in terms of π. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.) B. There is no solution.

A. x= StartFraction pi Over 12 EndFraction comma StartFraction 5 pi Over 12 EndFraction comma StartFraction 17 pi Over 12 EndFraction comma StartFraction 13 pi Over 12 EndFraction π/12,5π/12,17π/12,13π/12

1.2 If you have the graph of y=​f(x), how do you obtain the graph of y=​f(x+6​)? Question content area bottom Part 1 Choose the correct answer below. A. Shift the graph to the right 6 units. B. Shift the graph to the left 6 units. C. Shift the graph down 6 units. D. Shift the graph up 6 units.

B. Shift the graph to the left 6 units. Note: Recall that the graph y=​f(x−​b) is the graph of y=​f(x) shifted horizontally by b units​ (right if b>0 and left if b<​0). The graph of y=​f(x)+d is the graph of y=​f(x) shifted vertically by d units​ (up if d>0 and down if d<​0). ​Also, for c>​0, the graph of y=​cf(x) is the graph of y=​f(x) scaled vertically by a factor of c​ (broadened if 0<c<1 and steepened if c>​1).

1.3 Solve the equation. sin2x=1​, 0≤x<2π Question content area bottom Part 1 Choose the correct answer below. A. x=π6,5π6,3π2 B. x=π4,5π8 C. x=π4,5π4 D. x=π6,2π3,5π6

C. x=π/4,5π/4

1.3 What are the three Pythagorean identities for the trigonometric​ functions? Question content area bottom Part 1 Choose the correct answer below. A. tan2θ+1=sec2θ​, sin2θ=2sinθcosθ​, and cos2θ=cos2θ−sin2θ B. 1+cot2θ=csc2θ​, 2cos2θ=1+cos2θ​, and 2sin2θ=1−cos2θ C. sin2θ=2sinθcosθ​, sin2θ+cos2θ=1​, and cos2θ=cos2θ−sin2θ D. sin2θ+cos2θ=1​, 1+cot2θ=csc2θ​, and tan2θ+1=sec2θ

D. sin2θ+cos2θ=1​, 1+cot2θ=csc2θ​, and tan2θ+1=sec2θ

1.1 Does the independent variable of a function belong to the domain or​ range? Does the dependent variable belong to the domain or​ range? The independent variable of a function belongs to the ______The dependent variable of a function belongs to the _______

domain, range.

1.3 Define the six trigonometric functions in terms of the sides of a right triangle.

. sinθ= opposite side hypotenuse Part 2 Choose the correct answer below. cosθ= adjacent side hypotenuse Part 3 Choose the answers below. tanθ= opposite side adjacent side Part 4 Choose the correct answer below. cotθ= adjacent side opposite side Part 5 Choose the correct answer below. secθ= hypotenuse adjacent side Part 6 Choose the correct answer below. cscθ= hypotenuse opposite side

1.3 A light block hangs at rest from the end of a spring when it is pulled down 2 cm and released. Assume the block oscillates with an amplitude of 2 cm on either side of its initial​ position, and with a period of 8 s. Find a function d(t) that gives the displacement of the block t seconds after it is​ released, where d(t)>0 represents downward displacement. Question content area bottom Part 1 d(t)=____cos____ ​(Simplify your answers. Type exact​ answers, using π as needed. Use angle measures greater than or equal to 0 and less than 2π​.)

d(t)=2cos(π/4 t)

1.3 Design a sine function with the given properties. It has a period of 24 with a minimum value of 10 at t=3 and a maximum value of 16 at t=15. Question content area bottom Part 1 y=______ ​(Type an expression using t as the variable. Type an exact​ answer, using π as needed. Use integers or fractions for any numbers in the​ expression.)

y =3 sine left parenthesis StartFraction pi t Over 12 EndFraction minus StartFraction 3 pi Over 4 EndFraction right parenthesis plus 13 3sin(πt/12−3π/4)+13

1.1 example Decide whether graphs​ A, B, or both represent functions. xyAB A coordinate system has a horizontal x-axis and a vertical y-axis. From left to right, a curve labeled "A" starts at a minimum, rises to a maximum on the y-axis, and falls to a minimum. A U-shaped curve labeled "B" opens to the right and has vertex (0, 0).

A graph represents a function if and only if it passes the vertical line test. A graph passes the vertical line test if every vertical line intersects the graph at most once. A graph that fails this test does not represent a function. Part 2 Begin by examining graph A. Draw vertical lines through the graph. Notice that the vertical lines do not intersect the graph more than once. Any vertical line that can be drawn will only intersect graph A at most once. xyA A coordinate system has a horizontal x-axis and a vertical y-axis. From left to right, a curve labeled "A" starts at a minimum, rises to a maximum on the y-axis, and falls to a minimum. A vertical line intersects the curve at a point in quadrant 2. A second vertical line intersects the curve at a point in quadrant 1. Part 3 Since the graph passes the vertical line​ test, graph A represents a function. Part 4 Now examine graph B. Draw a vertical line through the graph. Notice that the vertical line intersects the graph twice. xyB A coordinate system has a horizontal x-axis and a vertical y-axis. A U-shaped curve labeled "B" opens to the right and has vertex (0, 0). A vertical line crosses the curve at a point in quadrant 1 and at a point in quadrant 4. Part 5 Since the graph does not pass the vertical line​ test, graph B does not represent a function. Part 6 ​Thus, only graph A represents the graph of a function. Note: A graph represents a function if and only if it passes the vertical line test. A graph passes the vertical line test if every vertical line intersects the graph at most once. A graph that fails this test does not represent a function.

1.2 If you have the graph of y=​f(x), how do you obtain the graph of y=​f(x+18​)? Question content area bottom Part 1 Choose the correct answer below. A. Shift the graph up 18 units. B. Shift the graph to the right 18 units. C. Shift the graph to the left 18 units. D. Shift the graph down 18 units.

C. Shift the graph to the left 18 units.

1.3 Identify the amplitude and period of the following function. f(θ)=4sinπθ4 Question content area bottom Part 1 The amplitude is__ ​(Type an integer or a​ decimal.) Part 2 The period is __. ​(Type an integer or a​ decimal.)

Part 1 The amplitude is 4. ​(Type an integer or a​ decimal.) Part 2 The period is 8. ​(Type an integer or a​ decimal.)

1.3 example Solve the equation on the interval 0≤θ<2π. 8sin2θ=4 Question content area bottom

First, solve the equation for sin2θ. 8sin2θ = 4 sin2θ = 12 Divide both sides by 8 and simplify. Part 2 sinθ = ±22 Apply the square root method. Part 3 The period of the sine function is 2π. ​Thus, in the interval [0,2π), there are two angles θ for which sinθ=22 and two angles θ for which sinθ=−22. The angles are θ=π4​, θ=3π4​, θ=5π4​, and θ=7π4. Part 4 In the interval [0,2π), the solutions of 8sin2θ=4 are θ=π4​, θ=3π4​, θ=5π4​, and θ=7π4.

1.2 example Find the​ point(s) of intersection of the parabolas y=x2 and y=−x2+42x using analytical methods. Use a graphing utility only to check your work.

Set the functions equal to each other and solve for x to find the​ x-coordinate(s) of the​ point(s) of intersection using analytical methods. Part 2 Set the functions equal to each other. To solve for​ x, write the equation in the form ax2+bx+c=0. x2 = −x2+42x 2x2−42x = 0 Part 3 Factor the expression. 2x2−42x = 0 2x(x−21) = 0 Part 4 ​Therefore, the​ x-coordinates of the points of intersection are x=0 and x=21. Part 5 While either original function could be used to find the​ y-coordinates, find the​ y-coordinates of the points of intersection by substituting the​ x-coordinates into the function y=x2. Start with x=0. y = x2 = (0)2 = 0 Part 6 Now find the​ y-coordinate of the point of intersection where x=21. y = x2 = (21)2 = 441 Part 7 ​Therefore, the points of intersection are​ (0,0) and ​(21​,441​). The​ "intersect" function on a graphing utility can be used to verify the points of intersection.

1.1 Find possible choices for outer and inner functions f and g such that the given function h equals f◦g. Give the domain of h. h(x)=x7−43 Question content area bottom What are the functions​ f(x) and​ g(x)? The domain of h is ___

What are the functions​ f(x) and​ g(x)? ​f(x)=x cubedx3 ​g(x)=x Superscript 7 Baseline minus 4x7−4 The domain of h is (−∞,∞). ​(Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as​ needed.)

1.3 Given the following​ information, evaluate the other five functions. secθ=54 and 3π2<θ<2π

Part 1 cosθ=four fifths 4/5 Part 2 sinθ=negative three fifths −3/5 Part 3 tanθ=negative three fourths− 3/4 Part 4 cscθ=negative five thirds −5/3 Part 5 cotθ=negative four thirds −4/3

1.3 example A projectile is launched from the ground at an angle of θ above the horizontal with an initial speed of v in​ ft/s. The range​ (the total distance traveled by the projectile over level​ ground) of the projectile is approximated by the equation x=v232sin2θ. Find the launch angle of a projectile with an initial speed of 85​ft/s and a range of 170 ft.

Using the formula for​ x, let v=85​, and x=170. x = v232sin2θ 170 = (85)232sin2θ Part 2 Solve for sin(2θ). 170 = (85)2sin(2θ)32 5440 = 7225sin(2θ) Evaluate the exponent and multiply both sides by 32. 0.7529 ≈ sin(2θ) Divide both sides by 7225. Part 3 Determine the two angles where sin(2θ)=0.7529. sin(2θ) ≈ 0.7529 2θ ≈ sin−1(0.7529) Part 4 Find the values for θ​, rounding to the nearest degree. 2θ ≈ 48.84° or 2θ ≈ 180o−48.84° 2θ ≈ 48.84° 2θ ≈ 131.16° θ ≈ 24° θ ≈ 66° Divide both sides by 2. Part 5 ​Therefore, the projectile will travel 170 feet if it is launched with an angle of elevation of approximately 24° or 66° and an initial velocity of 85 feet per second.

1.2 A club plans to have a fundraiser for which ​$8 tickets will be sold. The cost of room rental and refreshments is ​$140. Find and graph the function p=​f(n) that gives the profit from the fundraiser when n tickets are sold. Notice that ​f(0)=−​$140​; that​ is, the cost of room rental and refreshments must be paid regardless of how many tickets are sold. How many tickets must be sold to break even​ (zero profit)?

What is the function that gives the profit p from the fundraiser when n tickets are​ sold? p=8 n minus 140 8n−140 ​(Use integers or decimals for any numbers in the​ equation.) Part 2 Use the graphing tool to graph the function. Part 3 How many tickets must be sold to break even​ (zero profit)? 18 ​(Round up to the nearest​ ticket.)

1.3 Solve the equation on the interval 0≤θ<2π. 12sin2θ=3 Question content area bottom Part 1 What are the solutions in the interval 0≤θ<2π?

θ= StartFraction pi Over 6 EndFraction comma StartFraction 5 pi Over 6 EndFraction comma StartFraction 7 pi Over 6 EndFraction comma StartFraction 11 pi Over 6 EndFraction π/6,5π/6,7π/6,11π/6 ​(Simplify your answer. Type an exact​ answer, using π as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)

1.3 A projectile is launched from the ground at an angle of θ above the horizontal with an initial speed of v in​ ft/s. The range​ (the total distance traveled by the projectile over level​ ground) of the projectile is approximated by the equation x=v232sin2θ. Find the launch angle of a projectile with an initial speed of 80​ft/s and a range of 170 ft. Question content area bottom Part 1 θ≈___ degrees ​(Type your answer in degrees. Use a comma to separate answers as needed. Round to the nearest integer as​ needed.)

θ≈61,29

1.2 Determine the slope function for the function on the right. A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 2 to 7 in increments of 1. A graph consists of two rays. The first ray falls from right to left, starting at the point (0, 3) and passing through the point (negative 1, 2). The second ray falls from left to right, starting at the point (0, 3) and passing through the point (5, 1). Choose the correct slope function below.

1 if x<0 -2/5 if x>0 Note: Find the slope of the left piece of the​ graph, and then find the slope of the right piece of the graph. The slope of a line which goes through the points x1,y1 and x2,y2 is m=y2−y1x2−x1. Use the two slopes to define the slope​ function, if the slope function exists.

1.2 Determine the slope function for the function on the right. A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 2 to 7 in increments of 1. A graph consists of two rays. The first ray falls from right to left, starting at the point (0, 3) and passing through the point (negative 1, 2). The second ray falls from left to right, starting at the point (0, 3) and passing through the point (3, 1). Question content area bottom Part 1 Choose the correct slope function below. A. ​s(x)= enter your response here if x<0 enter your response here if x>0 B. ​s(x)=enter your response here C. The slope function is undefined.

A. ​s(x)= 1 if x<0 −2/3 if x>0

1.1 Explain how to find the domain of f◦g if you know the domain and range of f and g. Question content area bottom Part 1 Choose the correct answer below. A. The domain of f◦g consists of all x in the domain of g. B. Identify the​ x-values in the domain of g whose corresponding range values are in the domain of f. C. The domain of f◦g consists of all x in the domain of f. D. Identify the​ x-values in the domain of f whose corresponding range values are in the domain of g.

B. Identify the​ x-values in the domain of g whose corresponding range values are in the domain of f.

1.1 Determine whether the graph of the following function has symmetry about the​ x-axis, the​ y-axis, or the origin. Check your work by graphing. f(x)=2x5+2x3+3 Question content area bottom Part 1 S

D. The function has no symmetry with respect to the​ x-axis, y-axis, or origin.

1.2 Use shifts and scalings to transform the graph of f(x)=x into the graph of g. Use a graphing utility to check your work. a. g(x)=f(x+2) b. g(x)=2f(2x−3) c. g(x)=x−3 d. g(x)=2x−3−5 Part 1 a. Which transformation is used to transform the graph of f into the graph of​ g? A. Shift 2 units up. B. Shift 2 units to the right. C. Scale vertically by a factor of 2. D. Shift 2 units to the left. Your answer is correct. Part 2 Graph the function. Choose the correct graph below. A. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (0, 0) and passes through the points (1, 2) and (4, 4). B. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (0, 2) and passes through the points (4, 4) and (9, 5). C. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (2, 0) and passes through the points (3, 1) and (6, 2). D. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (negative 2, 0) and passes through the points (2, 2) and (7, 3). Part 3 b. Which transformations are used to transform the graph of f into the graph of​ g? A. Scale horizontally by a factor of 12​, then shift 32 units to the right. Then scale vertically by a factor of 2. B. Scale horizontally by a factor of 12​, then shift 3 units to the left. Then scale vertically by a factor of 2. C. Scale horizontally by a factor of 12 and reflect across the​ y-axis, then shift 3 units to the right. Then scale vertically by a factor of 2. D. Scale horizontally by a factor of 12​, then shift 3 units to the left. Then scale vertically by a factor of 12. Graph the function. Choose the correct graph below. A. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (1.5, 0) and passes through the points (5, 5.3) and (8, 7.2). All coordinates are approximate. B. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (negative 1.5, 0) and passes through the points (5, 1.8) and (8, 2.2). All coordinates are approximate. C. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from right to left at a decreasing rate starts at the point (negative 1.5, 0) and passes through the points (negative 5, 5.3) and (negative 8, 7.2). All coordinates are approximate. D. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (negative 1.5, 0) and passes through the points (5, 7.2) and (8, 8.7). All coordinates are approximate. Part 5 c. Which​ transformation(s) is/are used to transform the graph of f into the graph of​ g? A. Shift 3 units down. B. Shift 3 units to the left. C. Scale horizontally by a factor of 3 and reflect across the​ y-axis. D. Shift 3 units to the right. Graph the function. Choose the correct graph below. A. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (negative 3, 0) and passes through the points (1, 2) and (6, 3). B. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from right to left at a decreasing rate starts at the point (0, 0) and passes through the points (negative 3, 3) and (negative 10, 5.5). All coordinates are approximate. C. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (3, 0) and passes through the points (7, 2) and (12, 3). D. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (0, negative 3) and passes through the points (4, negative 1) and (9, 0). Part 7 d. Which transformations are used to transform the graph of f into the graph of​ g? A. Shift 3 units to the​ right, then scale vertically by a factor of 2. Then shift 5 units down. B. Scale horizontally by a factor of 2 and reflect across the​ y-axis, then shift 3 units to the right. Then shift 5 units down. C. Shift 3 units to the​ right, then scale vertically by a factor of 12. Then shift 5 units down. D. Shift 3 units to the​ left, then scale vertically by a factor of 2. Then shift 5 units up.

D. Shift 2 units to the left. D Graph A. Scale horizontally by a factor of 12​, then shift 32 units to the right. Then scale vertically by a factor of 2. A. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (1.5, 0) and passes through the points (5, 5.3) and (8, 7.2). All coordinates are approximate. D. Shift 3 units to the right. C. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve that rises from left to right at a decreasing rate starts at the point (3, 0) and passes through the points (7, 2) and (12, 3). A. Shift 3 units to the​ right, then scale vertically by a factor of 2. Then shift 5 units down.

1.2 example Graph the function. f(x)=x2−5xx−5if x≠58if x=5 Question content area bottom Part 1 G

Graph the function as two parts. The first part of the function is ​f(x)=x2−5xx−5 for x≠5. When x≠5​, the denominator is​ nonzero, so it can be simplified. Part 2 Factor the numerator. ​f(x)=x2−5xx−5​, x≠5 ​f(x)=x(x−5)x−5​, x≠5 Part 3 Simplify the expression by dividing out common factors. ​f(x)=x(x−5)x−5​, x≠5 ​f(x)=​x, x≠5 Part 4 So ​f(x)=x for x≠5. Since x≠5​, there must be a hole at the point ​(5​,5​). Part 5 The second part of the function ​f(x)=8 for x=5. ​So, the second part of the function consists of the point (5,8). Part 6 The graph of the piecewise function is shown to the right. f(x)=x2−5xx−5if x≠58if x=5 A coordinate system with a horizontal x-axis labeled from negative 10 to 10 in increments of 2, a vertical y-axis labeled from negative 10 to 10 in increments of 2. A graph has two parts. The first part is a line that rises from left to right passing through (negative 1,negative 1) and (0,0) with an open circle at (5,5). The second part is a point at (5,8).

A pole of length L is carried horizontally around a corner where a 2​-ft-wide hallway meets a 7​-ft-wide ​hallway, as shown in the figure on the right. For 0<θ<π2​, find the relationship between L and θ at the moment when the pole simultaneously touches both walls and the corner P. Estimate θ when L=13 ft. 2 ft 7 ft Question content area bottom Part 1 Identify the relationship between L and θ when the pole simultaneously touches both walls and the corner P. Choose the correct answer below. A. L(θ)=2sinθ+7cosθ B. L(θ)=2secθ+7cscθ C. L(θ)=2cosθ+7sinθ D. L(θ)=2cscθ+7secθ When L=14 ​ft, θ=___ ​(Round to two decimal places as needed. Use a comma to separate answers as​ needed.)

Part 1 A. L(θ)=2sinθ+7cosθ Note: Let​ P(x,y) be a point on a circle with radius r associated with angle θ. Then each of the trigonometric equations below is true. Use one of these functions to determine the length L as a function of θ. Notice that the length L is the sum of the lengths of the hypotenuses of the two right triangles formed by the​ pole, the widths of the​ hallways, and the corner P. sinθ=opp/hyp cosθ=adj/hyp tanθ=opp/adj cotθ=adj/opp secθ=hyp/adj cscθ=hyp/opp Part 2 Note:To solve for θ​, graph L(θ) and the given length of the pole L on the same coordinate axes. Then find the intersection point to find the​ value(s) of θ. 0.57, 1.07

1.1 example If ​f(x)=7/x^2+2​, what is ​f(5​)? What is f(t^2)​?

The argument of a function is the expression on which the function works. The argument in ​f(5​) is 5. Part 2 ​Now, evaluate ​f(5​). Replace x in ​f(x)=7x2+2 with the argument. ​f(x) = 7x2+2 ​f(5​) = 752+2 Substitute the argument for x. Part 3 Evaluate. ​f(5​) = 752+2 = 727 Simplify. Part 4 ​Now, evaluate ft2. The argument in ft2 is t2. Part 5 Replace x in ​f(x)=7x2+2 with the argument. ​f(x) = 7x2+2 ft2 = 7t22+2 Substitute the argument for x. Part 6 Evaluate. ​f(t2​) = 7t22+2 = 7t4+2 Simplify.

1.3 example Prove that the following equation is an identity. csc(θ)/sin(θ)−cot(θ)/tan(θ)=1

To prove that the given equation is an​ identity, first simplify the left side of the equation. Part 2 Use reciprocal identities and simplify. csc(θ)sin(θ)−cot(θ)tan(θ) =csc(θ)•1sin(θ)−cot(θ)•1tan(θ) =csc2(θ)−cot2(θ) Part 3 Use the Pythagorean identity. csc(θ)sin(θ)−cot(θ)tan(θ) =csc2(θ)−cot2(θ) =1 Part 4 ​Therefore, csc(θ)sin(θ)−cot(θ)tan(θ)=1.

1.3 example ​Solve, finding all solutions. tanx=sprt3/3 Question content area bottom

To​ begin, find a solution to tanx=33 between 0 and π radians on the unit circle. Because tanπ6=33​, one solution is x=π6. Part 2 Since you need to find all​ solutions, add integer multiples of the period of the tangent function. The period of the tangent function is π. Part 3 ​Thus, the solution to the equation tanx=33 is x=π6+π​k, where k is any integer.

1.2 Use shifts and scalings to graph the given function. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. p(x)=x2+5x−1

What is the original​ function? y=x squaredx2 ​(Type an expression. Simplify your​ answer.) Part 2 Choose the correct graph of​ p(x) below. c. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A parabola opens upwards, has vertex (negative 2.5, negative 7.25), and passes through the points (negative 3.5, negative 6.25) and (negative 1.5, negative 6.25).

1.2 Suppose the probability of a server winning any given point in a tennis match is a constant​ p, with 0≤p≤1. Then the probability of the server winning a game when serving from deuce is given below. Complete parts a and b below. f(p)=p21−2p(1−p) Part 1 a. Evaluate ​f(0.78​) and interpret the results. ​f(0.78​)=enter your response here ​(Round to four decimal places as​ needed.) Part 2 Interpret the results. When serving from​ deuce, the server will win approximately enter your response here out of 100 games. ​(Round to the nearest integer as​ needed.)

a. Evaluate ​f(0.78​) and interpret the results. ​f(0.78​)=0.9263 ​(Round to four decimal places as​ needed.) Part 2 Interpret the results. When serving from​ deuce, the server will win approximately 93 out of 100 games. ​(Round to the nearest integer as​ needed.) Part 3 b. Evaluate ​f(0.22​) and interpret the results. ​f(0.22​)=0.0737 ​(Round to four decimal places as​ needed.) Part 4 Interpret the results. When serving from​ deuce, the server will win approximately 7 out of 100 games. ​(Round to the nearest integer as​ needed.)

1.2 Consider the function ​f(x)=x3−3x2+12. a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the​ function, such as​ peaks, valleys, and intercepts.

a. Choose the correct graph below. The graphs are shown in a ​[−​8,8,2] by ​[−​16,16,2] window. A. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,negative 4), (0,0), (1,negative 2), (2,negative 4), and (2.5,negative 3.1). All points are approximate. B. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 2.5,negative 8.9), (negative 2,negative 8), (negative 1,negative 10), (0,negative 12), and (1,negative 8). All points are approximate. C. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,8), (0,12), (1,10), (2,8), and (2.5,8.9). All points are approximate. Your answer is correct. D. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve falls at a decreasing rate to a local minimum, then begins to rise to a local maximum, then begins to fall at an decreasing rate, passing through the points (negative 2.5,8.9), (negative 2,8), (negative 1,10), (0,12), and (1,8). All points are approximate. Part 2 b. Find the domain of the function. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The domain is restricted to the interval enter your response here. ​(Simplify your answer. Type your answer in interval​ notation.) B. The domain is the set of all real numbers. Your answer is correct. Part 3 c. Use a graphing utility to determine the peaks of the function. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The function has​ peak(s) at or near x=0. ​(Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as​ needed.) Your answer is correct. B. The function has no peak. Part 4 Use a graphing utility to determine the valleys of the function. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The function has​ valley(s) at or near x=2. ​(Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as​ needed.) Your answer is correct. B. The function has no valley. Part 5 Find intercepts of the function. Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice. ​(Type an ordered​ pair, using integers or decimals. Round to two decimal places as needed. Use a comma to separate answers as​ needed.) A. The​ x-intercept(s) is/are left parenthesis negative 1.61 comma 0 right parenthesis(−1.61,0) and the​ y-intercept(s) is/are left parenthesis 0 comma 12 right parenthesis(0,12). Your answer is correct. B. There is no​ x-intercept and the​ y-intercept(s) is/are enter your response here. C. The​ x-intercept(s) is/are enter your response here and there is no​ y-intercept. D. There is no​ x-intercept and there is no​ y-intercept.

1.1 example Evaluate each expression using the graphs of y=f(x) and y=g(x) shown below. a. f(g(3)) b. g(f(1)) c. f(g(5)) d. g(f(6)) e. f(f(1)) f. g(f(g(9)))

a. To evaluate f(g(3))​, first find g(3). Look at the graph to find the value of g when x=3. g(3)=3 Part 2 ​Next, substitute 3 for g(3) in f(g(3)). f(g(3))=​f(3​) Part 3 ​Finally, to evaluate ​f(3​). Look at the graph to find the value of f when x=3. f(g(3))=​f(3​)=6 Part 4 b. Evaluate g(f(1)) in a similar way. First find ​f(1​). ​f(1​)=3 Part 5 Now substitute 3 for ​f(1​) in g(f(1)). Then find the value of g when x=3. g(f(1)) = ​g(3​) = 3 Part 6 c. Use the same process to evaluate f(g(5)). First find ​g(5​). ​g(5​)=2 Part 7 Now substitute 2 for ​g(5​) in f(g(5)). Then find the value of f when x=2. f(g(5)) = ​f(2​) = 4 Part 8 d. Evaluate g(f(6)). First find ​f(6​). ​f(6​)=6 Part 9 Now substitute 6 for ​f(6​) in g(f(6)). Then find the value of g when x=6. g(f(6)) = ​g(6​) = 3 Part 10 e. Evaluate f(f(1)). First find ​f(1​). ​f(1​)=3 Part 11 Now substitute 3 for f in f(f(1)). Then find the value of f when x=3. f(f(1)) = ​f(3​) = 6 Part 12 f. Evaluate g(f(g(9))). First find ​g(9​). ​g(9​)=6 Part 13 Now substitute 6 for ​g(9​) in f(g(9)). Then find the value of f when x=6. f(g(9)) = ​f(6​) = 6 Part 14 ​Finally, substitute 6 for f(g(9)) in g(f(g(9))). Then find the value of g when x=6. g(f(g(9))) = ​g(6​) = 3

1.1 Use the terms​ domain, range, independent​ variable, and dependent variable to explain how a function relates one variable to another variable. Question content area bottom Part 1 Fill in the blanks below. A function is a rule that assigns to each value of the _______ in the ________ a unique value of the__________ in the ▼_____.

independent variable, domain, dependent variable​, range. A function is a rule that assigns to each value of the independent variable in the domain a unique value of the dependent variable in the range. Note: Recall that a function f is a rule that assigns to each value x in a set D a unique value denoted​ f(x). The set D is the domain of the function. The range is the set of all values of​ f(x) produced as x varies over the domain. The independent variable is the variable associated with the​ domain; the dependent variable belongs to the range.

1.2 Use shifts and scalings to graph the given function. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. p(x)=x2+5x0 Part 1 What is the original​ function? y= ​(Type an expression. Simplify your​ answer.) Part 2 Choose the correct graph of​ p(x) below. A. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A parabola opens upwards, has vertex (negative 2.5, negative 6.25), and passes through the points (negative 3.5, negative 5.25) and (negative 1.5, negative 5.25). B. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A parabola opens upwards, has vertex (6.25, 2.5), and passes through the points (5.25, 3.5) and (7.25, 3.5). C. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A line rises from left to right and passes through the points (0, negative 8.75) and (8.75, 0). D. -1010-1010xy

y=x squaredx2 A. -1010-1010xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A parabola opens upwards, has vertex (negative 2.5, negative 6.25), and passes through the points (negative 3.5, negative 5.25) and (negative 1.5, negative 5.25). Note: Apply shifts and scalings to the original function found in the previous step to graph the given function. Given the real numbers​ a, b,​ c, and d and the function​ f, the graph of y=cf(a(x−b))+d is obtained from the graph of y=​f(x) in the following steps. horizontal scaling by a factor of ​1/a y = f(x) y = ​f(ax) horizontal shift by b units y = ​f(a(x−​b)) vertical scaling by a factor of c y = ​cf(a(x−​b)) vertical shift by d units y = ​cf(a(x−​b))+d OK

1.2 A taxicab ride costs ​$4 plus ​$1.50 per mile. Let m be the distance​ (in miles) from the airport to a hotel. Find and graph the function​ c(m) that represents the cost of taking a taxi from the airport to the hotel. Also determine how much it costs if the hotel is 8 miles from the airport. Question content area bottom left Part 1 ​c(m)=enter your response here

​c(m)=1.5 m plus 4 1.5m+4 Part 2 Use the graphing tool to graph the function. Part 3 The taxicab costs ​$16 if the hotel is 8 miles from the airport.

1.2 Determine whether the following statements are true and give an explanation or counterexample. a. All polynomials are rational functions but not all rational functions are polynomials. b. If f is a linear​ polynomial, then f◦f is a quadratic polynomial. c. If f and g are​ polynomials, then the degrees of f◦g and g◦f are equal. d. To graph ​g(x)=​f(x+​2), shift the graph of f two units to the right. Question content area bottom Part 1 a. Choose the correct answer below. A. The statement is true because both polynomial functions and rational functions are algebraic functions. B. The statement is true. A polynomial function​ p(x) can be expressed as ​f(x)=p(x)1​, making it a rational function because ​q(x)=1 is a polynomial function.​ However, since rational functions are of the form p(x)q(x)​, where p and q are​ polynomials, not all rational functions are polynomial functions. For​ example, ​f(x)=x+1x is a rational​ function, but is not a polynomial function. C. The statement is false. Let ​p(x)=2x3+4x2 and ​q(x)=1. If ​f(x)=p(x)q(x)​, then​ f(x) is a rational function that is also a polynomial function. . Part 2 b. Choose the correct answer below. A. The statement is true. The degree of the composition function f◦g is the sum or the degrees of the functions f and g. If both functions are​ linear, the sum of their degrees is 2. B. The statement is true. Let ​f(x)=ax+b be a linear function where a and b are constants. Then f◦f=​(ax+​b)(ax+​b)=a2x2+2abx+b2. C. The statement is false. Let ​f(x)=x+2. Then f◦f=​f(f(x))=​(x+​2)+2=x+4. This is a linear function. Part 3 c. Choose the correct answer below. A. The statement is true. Let ​f(x)=anxn+an−1xn−1+⋯+a1x+a0​, a degree n​ polynomial, and ​g(x)=bnxm+bn−1xm−1+⋯+b1x+b0​, a degree m polynomial. The first term of f◦​g(x) is anbnmxn•m and the first term of g◦​f(x) is bmamnxm•n. ​Thus, the degree of f◦​g(x), n•​m, is equal to the degree of g◦​f(x), m•n. B. The statement is true. Since f◦g=g◦f for any f and​ g, their degrees must be the same. C. The statement is false. Let f=1x and ​g(x)=x2+x. Then f◦g=1x2+x which has a degree of −2. ​However, g◦f=x+​1, which has a degree of 1. Part 4 d. Choose the correct answer below. A. The statement is true. The graph of y=​f(x)+d is the graph of y=​f(x) shifted horizontally by d units​ (right if d>0 and down if d<​0). Since b=2>​0, the graph is shifted to the right. B. The statement is true. The graph of y=​f(x−​b) is the graph of y=​f(x) shifted horizontally by b units​ (right if b<0 and left if b>​0). Since b=−2<​0, the graph is shifted to the right. C. The statement is false. The graph of y=​f(x−​b) is the graph of y=​f(x) shifted horizontally by b units​ (right if b>0 and left if b<​0). Since b=−2<​0, the graph is shifted to the left.

Part 1 a. Choose the correct answer below. B. The statement is true. A polynomial function​ p(x) can be expressed as ​f(x)=p(x)1​, making it a rational function because ​q(x)=1 is a polynomial function.​ However, since rational functions are of the form p(x)q(x)​, where p and q are​ polynomials, not all rational functions are polynomial functions. For​ example, ​f(x)=x+1x is a rational​ function, but is not a polynomial function. Part 2 b. Choose the correct answer below. C. The statement is false. Let ​f(x)=x+2. Then f◦f=​f(f(x))=​(x+​2)+2=x+4. This is a linear function. Part 3 c. Choose the correct answer below. A. The statement is true. Let ​f(x)=anxn+an−1xn−1+⋯+a1x+a0​, a degree n​ polynomial, and ​g(x)=bnxm+bn−1xm−1+⋯+b1x+b0​, a degree m polynomial. The first term of f◦​g(x) is anbnmxn•m and the first term of g◦​f(x) is bmamnxm•n. ​Thus, the degree of f◦​g(x), n•​m, is equal to the degree of g◦​f(x), m•n. Part 4 d. Choose the correct answer below. C. The statement is false. The graph of y=​f(x−​b) is the graph of y=​f(x) shifted horizontally by b units​ (right if b>0 and left if b<​0). Since b=−2<​0, the graph is shifted to the left.

1.3 example A light block hangs at rest from the end of a spring when it is pulled down 11 cm and released. Assume the block oscillates with an amplitude of 11 cm on either side of its initial​ position, and with a period of 5 s. Find a function d(t) that gives the displacement of the block t seconds after it is​ released, where d(t)>0 represents downward displacement. Question content area bottom Part 1

When the object is​ released, the displacement of the object from the rest position is 11 cm. Because d(t)=11 when t=0, use the following cosine function to describe the motion rather than a sine​ function, which would require more transformations. d(t)=Acos(Bt) Part 2 In d(t)=Acos(Bt)​, A represents the vertical stretch or amplitude and 2πB represents the period of the function. Part 3 Since the block oscillates with an amplitude of 11 cm on either side of its initial​ position, A=11. Part 4 Now use the period to solve for B. period = 2πB 5 = 2πB B = 2π5 Part 5 ​Therefore, given that A=11 and B=2π5​, the function that models the motion of the light block is d(t)=11cos2π5t.

1.3 A pole of length L is carried horizontally around a corner where a 2​-ft-wide hallway meets a 5​-ft-wide ​hallway, as shown in the figure on the right. For 0<θ<π2​, find the relationship between L and θ at the moment when the pole simultaneously touches both walls and the corner P. Estimate θ when L=11 ft. 2 ft 5 ft Question content area bottom Part 1 Identify the relationship between L and θ when the pole simultaneously touches both walls and the corner P. Choose the correct answer below. Part 2 When L=11 ​ft, θ=_____ ​(Round to two decimal places as needed. Use a comma to separate answers as​ needed.)

C. L(θ)=2secθ+5cscθ

1.2 What is the domain of a​ polynomial? Question content area bottom Choose the correct answer below. A. The domain of a polynomial is the set of all real numbers from one to infinity. B. The domain of a polynomial is the set of all rational numbers. C. The domain of a polynomial is the set of all real numbers. D. The domain of a polynomial is the set of all real numbers from zero to infinity.

C. The domain of a polynomial is the set of all real numbers. Note: Recall that polynomials are functions of the form f(x)=anxn+an−1xn−1+...+a1x+a0​, where the coefficients a0,a1,...,an are real numbers with an≠0 and the nonnegative integer n is the degree of the polynomial.

1.1 A function and an interval of its independent variable are given. The endpoints of the interval are associated with the points P and Q on the graph of the function. Answer parts a and b. The volume V of a gas in cubic centimeters is given by V=5p​, where p is the pressure in atmospheres and 0.5≤p≤2. a. Sketch a graph of the function and the secant line through P and Q. b. Find the slope of the secant line in part​ (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer. Question content area bottom Part 1 a. Choose the correct graph below. A. 01234024681012pV A coordinate system has a horizontal p-axis labeled from 0 to 4 in increments of 0.5 and a vertical V-axis labeled from 0 to 12 in increments of 1. From left to right, a curve falls at a decreasing rate, passing through the plotted points (0.25, 10) and (3, 0.8). From left to right, a line falls, intersecting the curve at the plotted points (0.25, 10) and (3, 0.8). All coordinates are approximate. B. 01234024681012pV A coordinate system has a horizontal p-axis labeled from 0 to 4 in increments of 0.5 and a vertical V-axis labeled from 0 to 12 in increments of 1. From left to right, a curve rises at an increasing rate, passing through the plotted points (1, 0.7) and (3, 3.5). From left to right, a line rises, intersecting the curve at the plotted points (1, 0.7) and (3, 3.5). All coordinates are approximate. C. 01234024681012pV A coordinate system has a horizontal p-axis labeled from 0 to 4 in increments of 0.5 and a vertical V-axis labeled from 0 to 12 in increments of 1. From left to right, a curve falls at a decreasing rate, passing through the plotted points (0.5, 10) and (2, 2.5). From left to right, a line falls, intersecting the curve at the plotted points (0.5, 10) and (2, 2.5). All coordinates are approximate. D. 01234024681012pV A coordinate system has a horizontal p-axis labeled from 0 to 4 in increments of 0.5 and a vertical V-axis labeled from 0 to 12 in increments of 1. From left to right, a curve rises at an increasing rate, passing through the plotted points (1, 1.4) and (3, 5.5). From left to right, a line rises, intersecting the curve at the plotted points (1, 1.4) and (3, 5.5). All coordinates are approximate. Part 2 b. The slope of the secant line is _____. This means the ____ changes at an average rate of ___ ____ over the interval 0.5≤p≤2. ​(Type integers or​ decimals.)

C. 01234024681012pV A coordinate system has a horizontal p-axis labeled from 0 to 4 in increments of 0.5 and a vertical V-axis labeled from 0 to 12 in increments of 1. From left to right, a curve falls at a decreasing rate, passing through the plotted points (0.5, 10) and (2, 2.5). From left to right, a line falls, intersecting the curve at the plotted points (0.5, 10) and (2, 2.5). All coordinates are approximate. The slope of the secant line is negative 5−5. This means the volume changes at an average rate of −5 cm cubed Over atmosphere over the interval 0.5≤p≤2.

1.2 Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 144 ft3. a. Find and graph the function​ S(x) that gives the surface area of the​ box, for all values of x>0. b. Based on your graph in part​ (a), estimate the value of x that produces the box with a minimum surface area. Question content area bottom Part 1 a. Determine the function​ S(x). A. S(x)=144x2 B. S(x)=x2+4x C. S(x)=x2+144x2 D. S(x)=x2+576x Graph the function​ S(x). Choose the correct graph below. The graphs are displayed using a window size of​ [0,25,1] by​ [0,250,10]. A. A coordinate system has a horizontal axis labeled from 0 to 25 in increments of 1 and a vertical axis labeled from 0 to 250 in increments of 10. From left to right, a curve falls steeply at a decreasing rate, passing through (1.5, 66) to a minimum at (3.5, 24), and then rises at an increasing rate, passing through (12, 145). All coordinates are approximate. B. A coordinate system has a horizontal axis labeled from 0 to 25 in increments of 1 and a vertical axis labeled from 0 to 250 in increments of 10. From left to right, a curve starts at (0, 0), then rises at an increasing rate, passing through (1.5, 8), (3.5, 26), and (12, 192). All coordinates are approximate. C. A coordinate system has a horizontal axis labeled from 0 to 25 in increments of 1 and a vertical axis labeled from 0 to 250 in increments of 10. From left to right, a curve falls steeply at a decreasing rate, passing through (3, 201) to a minimum at (6.6, 131), and then rises at an increasing rate, passing through (12, 192). All coordinates are approximate. D. A coordinate system has a horizontal axis labeled from 0 to 25 in increments of 1 and a vertical axis labeled from 0 to 250 in increments of 10. A curve is above the horizontal axis and to the right of the vertical axis, approaching both. b. x≈enter your response here ​(Round to two decimal places as​ needed.)

D. S(x)=x2+576x Note: To determine the surface area​ function, S(x), find a general function in terms of x and h for the surface area of the lidless box. Then use the given volume and the fact that the volume of the lidless box equals the area of the base times the height of the box to express the​ height, h, in terms of x. Substitute the value of h into the general surface area function to result in a surface area function in terms of x. C. A coordinate system has a horizontal axis labeled from 0 to 25 in increments of 1 and a vertical axis labeled from 0 to 250 in increments of 10. From left to right, a curve falls steeply at a decreasing rate, passing through (3, 201) to a minimum at (6.6, 131), and then rises at an increasing rate, passing through (12, 192). All coordinates are approximate. x=6.60 Note: Use either a graphing utility to estimate the value of x at the minimum of the graph or the​ by-hand method. To use the​ by-hand method, examine the previously created graph to determine the minimum of the graph. Estimate the value of x for the minimum of the graph to determine the value of x that results in the minimum surface area.

1.1 Determine an appropriate domain of the function. Identify the independent and dependent variables. A stone is thrown vertically upward from the ground at a speed of 40 ​m/s at time t=0. Its distance d​ (in m) above the ground​ (neglecting air​ resistance) is approximated by the function ​f(t)=40t−5t^2. Question content area bottom Part 1 An appropriate domain is____ ​(Simplify your answer. Type your answer in interval​ notation.)

An appropriate domain is t[0,8]. ​(Simplify your answer. Type your answer in interval​ notation.) Part 2 The independent variable is t and the dependent variable is d.

1.2 Given the graph of y=x2​, how do you obtain the graph y=2(x+8)2+3​? Question content area bottom Part 1 Choose the correct answer below. A. Shift the graph to the right 8 ​units, then compress the graph vertically by a factor of 2. Then shift the graph down 3 units. B. Shift the graph to the left 8 ​units, then stretch the graph vertically by a factor of 2. Then shift the graph up 3 units. C. Shift the graph to the right 8 ​units, then stretch the graph horizontally by a factor of 2. Then shift the graph up 3 units. D. Shift the graph to the left 2 ​units, then compress the graph horizontally by a factor of 8. Then shift the graph up 3 units.

B. Shift the graph to the left 8 ​units, then stretch the graph vertically by a factor of 2. Then shift the graph up 3 units.

1.1 A cylindrical water tower with a radius of 11 m and a height of 45 m is filled to a height of h. The volume V of water​ (in cubic​ meters) is given by the function​ g(h) = 121πh. Determine the appropriate domain of the function. Identify the independent and dependent variables. Question content area bottom Part 1 Select the correct choice​ below, and if​ necessary, fill in the box to complete your choice. ​(Simplify your answer. Type your answer in interval​ notation.) A. The independent variable is​ V; the dependent variable is h. The domain is D=enter your response here. B. The independent variable is​ h; the dependent variable is V. The domain is D=enter your response here.

B. The independent variable is​ h; the dependent variable is V. The domain is D=[0,45]. note: Determine which variable depends on the other. The independent variable is the variable associated with the domain while the dependent variable is associated with the range. The domain is all values of the independent variable for which the function is defined.

1.3 example Beginning with the graph of y=sinx​, use shifting and scaling transformations to sketch the graph of the following function. Use a graphing utility only to check your work. ​f(x)=7sin4x Question content area bottom left Part 1 Begin with the graph of the function ​f(x)=sinx​, shown to the right. ...

Begin with the graph of the function ​f(x)=sinx​, shown to the right. Part 2 The function y=Asin(B(θ−C))+D​, when compared to the graph of y=sinθ has an amplitude of A​, a period of 2πB​, a horizontal shift of​ C, and a vertical shift of D. Part 3 Determine the amplitude of the function ​f(x)=7sin4x. amplitude=7 Part 4 Using the results from the previous​ step, vertically stretch the graph of ​f(x)=sinx so that it has an amplitude of 7​, as shown to the right. Part 5 Next determine the period of the function ​f(x)=7sin4x. First determine B. B=4 Part 6 Using the results from the previous​ step, determine the period of ​f(x)=7sin4x. period = 2πB = 2π4 Substitute. Part 7 = π2 Simplify. Part 8 Horizontally scale the graph found in a previous step by 4​, so that the period of the new graph is π2​, as shown to the right. Part 9 Next determine the horizontal shift of the function ​f(x)=7sin4x. Recall that an equation of the form y=Asin(B(θ−C))+D has a horizontal shift of C. Determine the horizontal shift of the given function. C=0 Part 10 Since the value of the horizontal shift is​ 0, there is no horizontal shift. Part 11 ​Finally, determine the vertical shift of the function ​f(x)=7sin4x. Recall that an equation of the form y=Asin(B(θ−C))+D has a vertical shift of D. Determine the vertical shift of the given function. D=0 Part 12 Since the value of the vertical shift is​ 0, there is no vertical shift. Part 13 The graph of 7sin4x with amplitude 7 and period π2 is shown to the right.

1.2 Graph the square wave function defined as follows. f(x)= 0 if x<0 1 if 0≤x<1 0 if 1≤x<2 1 if 2≤x<3 Question content area bottom Part 1 Choose the correct graph below. A. -16-12xy A coordinate system has a horizontal x-axis labeled from negative 1 to 6 in increments of 1 and a vertical y-axis labeled from negative 1 to 2 in increments of 1. A graph consists of a horizontal ray pointing to the left from an open dot at (0, 1) and 5 horizontal line segments, each extending from a solid dot to an open dot. Their endpoints are listed from left to right as follows: (0, 0), (1, 0); (1, 1), (2, 1); (2, 0), (3, 0); (3, 1), (4, 1); (4, 0), (5, 0). B. -16-12xy A coordinate system has a horizontal x-axis labeled from negative 1 to 6 in increments of 1 and a vertical y-axis labeled from negative 1 to 2 in increments of 1. A graph consists of a horizontal ray pointing to the left from a solid dot at (0, 1) and 5 horizontal line segments, each extending from an open dot to a solid dot. Their endpoints are listed from left to right as follows: (0, 0), (1, 0); (1, 1), (2, 1); (2, 0), (3, 0); (3, 1), (4, 1); (4, 0), (5, 0). C. -16-12xy A coordinate system has a horizontal x-axis labeled from negative 1 to 6 in increments of 1 and a vertical y-axis labeled from negative 1 to 2 in increments of 1. A graph consists of a horizontal ray pointing to the left from a solid dot at (0, 0) and 5 horizontal line segments, each extending from an open dot to a solid dot. Their endpoints are listed from left to right as follows: (0, 1), (1, 1); (1, 0), (2, 0); (2, 1), (3, 1); (3, 0), (4, 0); (4, 1), (5, 1). D. -16-12xy A coordinate system has a horizontal x-axis labeled from negative 1 to 6 in increments of 1 and a vertical y-axis labeled from negative 1 to 2 in increments of 1. A graph consists of a horizontal ray pointing to the left from an open dot at (0, 0) and 5 horizontal line segments, each extending from a solid dot to an open dot. Their endpoints are listed from left to right as follows: (0, 1), (1, 1); (1, 0), (2, 0); (2, 1), (3, 1); (3, 0), (4, 0); (4, 1), (5, 1).

D. -16-12xy A coordinate system has a horizontal x-axis labeled from negative 1 to 6 in increments of 1 and a vertical y-axis labeled from negative 1 to 2 in increments of 1. A graph consists of a horizontal ray pointing to the left from an open dot at (0, 0) and 5 horizontal line segments, each extending from a solid dot to an open dot. Their endpoints are listed from left to right as follows: (0, 1), (1, 1); (1, 0), (2, 0); (2, 1), (3, 1); (3, 0), (4, 0); (4, 1), (5, 1).

1.2 example A club plans to have a fundraiser for which ​$11 tickets will be sold. The cost of room rental and refreshments is ​$150. Find and graph the function p=​f(n) that gives the profit from the fundraiser when n tickets are sold. Notice that ​f(0)=−​$150​; that​ is, the cost of room rental and refreshments must be paid regardless of how many tickets are sold. How many tickets must be sold to break even​ (zero profit)? Question content area bottom left Part 1 Find the constants m and b in the profit function p=​f(n)=mn+b.

Find the constants m and b in the profit function p=​f(n)=mn+b. Part 2 The slope represents the increase in profit for each additional ticket sold. The slope of the profit line is m=11. Part 3 The​ y-intercept represents the profit generated when no tickets are sold. The​ y-intercept of the profit line is ​(0,−150​). Part 4 Use the​ slope, m=11​, and​ y-intercept, ​(0,−150​), to write the function that gives the profit p from the fundraiser when n tickets are sold. p=11n−150 Part 5 Graph the profit function by plotting two points and drawing a line between them. The​ y-intercept, ​(0,−150​), is plotted to the right. Part 6 Use the function to find p when n=10. p = 11n−150 = 11(10)−150 Part 7 Simplify. p = 11(10)−150 = −40 Part 8 The point ​(10,−40​) is plotted to the right. Part 9 Draw a line between the points to graph the function. The function is plotted to the right. Part 10 Substitute p=0 into the function and solve for n to determine the number of tickets that must be sold to break even. p = 11n−150 0 = 11n−150 Part 11 Solve for n. Round up since only a whole number of tickets can be sold. 0 = 11n−150 150 = 11n 14 ≈ n Part 12 ​Therefore, 14 tickets must be sold to break even.

1.1 example A function and an interval of its independent variable are given. The endpoints of the interval are associated with the points P and Q on the graph of the function. Answer parts a and b. The volume V of a gas in cubic centimeters is given by V=16p​, where p is the pressure in atmospheres and 0.5≤p≤2. a. Sketch a graph of the function and the secant line through P and Q. b. Find the slope of the secant line in part​ (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

First, make a table using a few values of p. Find the corresponding values of V. p V 0.5 32 1 16 2 8 Part 2 List the points P and Q as ordered​ pairs, where P is the point at p=0.5 and Q is the point at p=2. P = ​(0.5,32​) Q = ​(2​,8​) Part 3 Plot the table points and connect them with a smooth curve. Then connect the points P and Q. The graph is shown to the right. 01234048121620242832pV A coordinate system has a horizontal p-axis labeled from 0 to 4 in increments of 0.5 and a vertical V-axis labeled from 0 to 34 in increments of 2. From left to right, a curve falls at a decreasing rate, passing through the plotted points (0.5, 32) and (2, 8). From left to right, a line falls, intersecting the curve at the plotted points (0.5, 32) and (2, 8). Part 4 b. Use the slope formula m=y2−y1x2−x1 to find the slope of the line that contains the points P and Q. Use the ordered pairs P=x1,y1 and Q=x2,y2 to find the​ slope, where x represents p and y represents V. Part 5 Find the slope. msec = y2−y1x2−x1 = 8−322−0.5 Substitute the values for the coordinates using P=x1,y1 and Q=x2,y2. = −16 Simplify. Part 6 Notice that x represents the pressure​ p, which is measured in​ atmospheres, and y represents the volume​ V, which is measured in cubic centimeters. The units for the slope are cm3atmosphere. Part 7 ​Therefore, the slope is −16 cm3atmosphere. The volume changes at an average rate of −16 cm3atmosphere over the interval 0.5≤p≤2.

1.2 example Write a definition of the function whose graph is given. A coordinate system has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 10 in increments of 1. A graph consists of two rays. One ray starts at an open dot at (3, 9) and falls from right to left, passing through the point (0, 6). A second ray falls from left to right, starting at a solid dot at (3, 2) and passing through the point (6, 1).

Functions that have different definitions on different parts of the domain are called piecewise functions. Notice that both pieces of the given function are linear. Part 2 First find where the function is defined for each linear piece. The piece on the left is a linear function defined for the domain x<3. Part 3 The piece on the right is a linear function defined for the domain x≥3. Part 4 Write the equation for the first linear function by finding the slope m and the​ y-intercept b of the line. Find the slope of the line on the left. m = change in ychange in x = 9−63−0 = 1 Part 5 The​ y-intercept b of the line on the left of the graph is 6. Part 6 Using y=mx+​b, the equation of the first line is y=x+6. Part 7 Write the equation for the second linear function by finding the​ y-intercept b and the slope m of the line. Find the slope of the line on the right. m = change in ychange in x = 0−29−3 = −13 Part 8 The value of b for the second line can be found by substituting the point ​(3,2​) and the slope −13 into the formula y=mx+b and solving for b. y = mx+b 2 = −13(3)+b b = 3 Part 9 Using y=mx+​b, the equation of the second line is y=−13x+3. Part 10 ​Now, use the information found above to define a piecewise function. The function definition is below. f(x)=x+6 if x<3 −13x+3 if x≥3

1.1 example If ​f(x)=x and ​g(x)=x3−28​, simplify the expressions (f◦g)(4)​, (f◦f)(100)​, (g◦f)(x)​, and (f◦g)(x).

Given two functions f and​ g, the composite function f◦g is defined by ​(f◦​g)(x)=​f(g(x)). It is evaluated in two steps as y=​f(u), where u=​g(x). Part 2 Let u=​g(4​) and evaluate ​g(4​). ​g(x) = x3−28 ​g(4​) = 43−28 = 36 Part 3 Now evaluate​ f(u) where u=36. ​f(x) = x ​f(36​) = 36 = 6 Part 4 ​Therefore, the expression (f◦g)(4) simplified is 6. Part 5 Evaluate (f◦f)(100) by letting y=​f(u), where u=​f(100​). ​First, evaluate ​f(100​). ​f(x) = x ​f(100​) = 100 = 10 Part 6 Now evaluate ​f(10​). ​f(x) = x ​f(10​) = 10 Part 7 ​Therefore, the expression (f◦f)(100) simplified is 10. Part 8 To simplify (g◦f)(x)​, substitute x as the argument of g. (g◦f)(x) = gx = x3−28 = x32−28 Part 9 ​Therefore, the expression (g◦f)(x) simplified is x32−28. Part 10 Use the same approach to find (f◦g)(x) by substituting x3−28 for the argument of f. (f◦g)(x) = fx3−28 = x3−28 Part 11 ​Therefore, the expression (f◦g)(x) simplified is x3−28.

1.1 example Assume f is an even function and g is an odd function. Assume f and g are defined for all real numbers. Use the table to evaluate the given compositions. x 1 2 3 4 ​f(x) 1 −2 3 −4 ​g(x) −4 −3 −2 −1 a. ​f(g(−​1)) b.​ g(f(−​4)) c. ​f(g(−​3)) d. ​f(g(−​2)) e. g​(g(−​1)) f. f​(g(0)−​1) g. f​(g(g(−​2))) h. g​(f(f(−​4))) i. g​(g(g(−​1)))

Given two functions f and​ g, the composite function f◦g is defined by ​(f◦​g)(x)=​f(g(x)). It is evaluated in two​ steps: y=​f(u), where u=​g(x). The domain of f◦g consists of all x in the domain of g such that u=​g(x) is in the domain of f. Part 2 An even function f has the property that ​f(−​x)=​f(x), for all x in the domain. The graph of an even function is symmetric about the​ y-axis. An odd function f has the property ​f(−​x)=−​f(x), for all x in the domain. The graph of an odd function is symmetric about the origin. Part 3 a. Use the table and the fact that g is an odd function to find ​g(−​1). ​g(−​1)=4 Part 4 This means that ​f(g(−​1))=​f(4​). Use the table to find ​f(4​). ​f(4​)=−4 Part 5 ​Thus, the value of the given composition is​ f(g(−​1))=−4. Part 6 b. Use the table and the fact that f is an even function to find ​f(−​4). ​f(−​4)=−4 Part 7 This means that ​g(f(−​4))=​g(−4​). Use the table and the fact that g is odd to find ​g(−4​). ​g(−4​)=1 Part 8 ​Thus, the value of the given composition is ​g(f(−​4))=1. Part 9 c. Use the table and the fact that g is an odd function to find ​g(−​3). ​g(−​3)=2 Part 10 This means that ​f(g(−​3))=​f(2​). Use the table to find ​f(2​). ​f(2​)=−2 Part 11 ​Thus, the value of the given composition is ​f(g(−​3))=−2. Part 12 d. Use the table and the fact that g is odd to find ​g(−​2). ​g(−​2)=3 Part 13 This means that ​f(g(−​2))=​f(3​). Use the table to find ​f(3​). ​f(3​)=3 Part 14 ​Thus, the value of the given composition is ​f(g(−​2))=3. Part 15 e. Use the table and the fact that g is odd to find ​g(−​1). ​g(−​1)=4 Part 16 This means that ​g(g(−​1))=​g(4​). Use the table to find ​g(4​). ​g(4​)=−1 Part 17 ​Thus, the value of the given composition is g​(g(−​1))=−1. Part 18 f. Use the fact that g is odd to find ​g(0)−1. ​g(0)−1=−1 Part 19 This means that ​f(g(0)−​1)=​f(−​1). Use the table to find ​f(−​1). ​f(−​1)=1 Part 20 ​Thus, the value of the given composition is f​(g(0)−​1)=1. Part 21 g. Use the table and the fact that g is odd to find ​g(−​2). ​g(−​2)=3 Part 22 This means that ​f(g(g(−​2)))=​f(g(3​)). Use the table to find ​g(3​). ​g(3​)=−2 Part 23 This means that ​f(g(g(−​2)))=​f(−2​). Use the table and the fact that f is even to find ​f(−2​). ​f(−2​)=−2 Part 24 ​Thus, the value of the given composition is f​(g(g(−​2)))=−2. Part 25 h. Use the table and the fact that f is even to find ​f(−​4). ​f(−​4)=−4 Part 26 This means that ​g(f(f(−​4)))=​g(f(−4​)). Use the table and the fact that f is even to find ​f(−4​). ​f(−4​)=−4 Part 27 This means that ​g(f(f(−​4)))=​g(−4​). Use the table and the fact that g is odd to find ​g(−4​). ​g(−4​)=1 Part 28 ​Thus, the value of the given composition is g​(f(f(−​4)))=1. Part 29 i. Use the table and the fact that g is odd to find ​g(−​1). ​g(−​1)=4 Part 30 This means that ​g(g(g(−​1)))=​g(g(4​)). Use the table to find ​g(4​). ​g(4​)= −1 Part 31 This means that ​g(g(g(−​1)))=​g(−1​). Use the table and the fact that g is odd to find ​g(−1​). ​g(−1​)=4 Part 32 ​Thus, the value of the given composition is ​g(g(g(−​1)))=4. Note:Given two functions f and​ g, the composite function f◦g is defined by ​(f◦​g)(x)=​f(g(x)). It is evaluated in two​ steps: y=​f(u), where u=​g(x). The domain of f◦g consists of all x in the domain of g such that u=​g(x) is in the domain of f. An even function f has the property that ​f(−​x)=​f(x), for all x in the domain. The graph of an even function is symmetric about the​ y-axis. An odd function f has the property ​f(−​x)=−​f(x), for all x in the domain. The graph of an odd function is symmetric about the origin.

1.3 Use the given figure to prove the Law of​ Sines, sinA/a=sinB/b=sinC/c. Question content area bottom Part 1 Let h be the altitude from vertex B to side b. Determine an expression for h in terms of the angle A and the length of side c.

Let h be the altitude from vertex B to side b. Determine an expression for h in terms of the angle A and the length of side c. h = c sine Upper AcsinA Part 2 Determine an expression for h in terms of the angle C and the length of side a. h =a sine Upper CasinC Part 3 Use the expressions previously found for h to write a new equation. sine Upper AsinAa =sine Upper CsinCc Part 4 Use similar reasoning to prove the other angles. Let h2 be the altitude from vertex C to side c. Determine an expression for h2 in terms of the angle B and the length of side a. h2 =a sine Upper BasinB A B C a b c h2 Part 5 Determine an expression for h2 in terms of the angle A and the length of side b. h2 = b sine Upper AbsinA Part 6 Use the expressions previously found for h2 to write a new equation. sine Upper AsinAa =sine Upper BsinBb A B C a b c h2 Part 7 ​Therefore, sine Upper AsinAa=sine Upper BsinBb=sine gCsinCc.

1.1 example The entire graph of f is given. State the domain and range of f. 02468100246810xyy=f(x) A coordinate plane has a horizontal x-axis labeled from 0 to 8 in increments of 1 and a vertical y-axis labeled from 0 to 8 in increments of 1. A downward-opening curve starts at the plotted point (5,7), rises to a maximum at (6,8), then falls ending at the plotted point (8,4). Point (5,7) is labeled as a closed circle and point (8,4) is labeled as an open circle.

The set of valid or meaningful inputs x is called the domain of the function. Part 2 From the​ graph, the domain of the function f is the set of all​ x-values for which the graph of f is defined. Part 3 Determine the values of x for which the graph of f is defined. ​{x | 5≤x<8​} Part 4 ​Therefore, the domain D of the function f is ​[5​,8​). Part 5 The range is the set of all values of​ f(x) produced as x varies over the entire domain. From the​ graph, the range of the function f is the set of all​ y-values that correspond to the​ x-values for which the graph of f is defined. Part 6 Determine the values of y for the corresponding values of x for which the graph of f is defined. ​{y | 4<y≤8​} Part 7 ​Therefore, the range R of the function f is ​(4​,8​].

1.3 Prove that the following equation is an identity. sin(c)/csc(c)+cos(c)/sec(c)=1

Simplify the left side of the given equation. sin(c)csc(c)+cos(c)sec(c) =sin(c)•1csc(c)+cos(c)•1sec(c) =sine squared left parenthesis c right parenthesis plus cosine squared left parenthesis c right parenthesis nbsp =sin2(c)+cos2(c) =1

1.2 example Determine the slope function​ S(x) for the following function. f(x)=l0.5xl

The slope​ function, S(x), is the slope of the curve​ f(x) at the point​ (x,f(x)). Part 2 First graph the function​ f(x). The graph of the function is shown to the right. -10-5510-10-5510xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A V shaped graph opens upward and passes through (negative 4,2), (negative 2,1), (0,0), (2,1), and (4,2). Part 3 Examine the graph of the function​ f(x). Notice that the slope changes at x=0. Part 4 The slope of​ f(x) is 0.5 for x>0. Part 5 The slope of​ f(x) is −0.5 for x<0. Part 6 ​Thus, the slope function is the piecewise function​ S(x) shown below. S(x)=0.5xif x>0−0.5xif x<0

1.3 example Without using a​ calculator, evaluate the following expression or state that the quantity is undefined. cot−14π3 Question content area bottom Part 1

The angle −14π3 is reached by rotating the positive​ x-axis in a clockwise direction. Part 2 The angle −14π3 corresponds to a clockwise revolution of 2 full​ circle(s) through 2π radians each plus an additional clockwise rotation through 2π3 radians. Part 3 In other​ words, −14π3=2​(−2π​)−2π3. ​Therefore, this angle has the same terminal side as an angle having a radian measure of 2π−2π3=4π3. Part 4 The coordinates of the point on the unit circle corresponding to the angle 4π3 are −12,−32. Part 5 A real number θ that determines a point​ (x,y) on the unit circle has the basic circular function cotθ=cosθsinθ​, where sinθ is the​ y-coordinate of the point on the unit​ circle, and cosθ is the​ x-coordinate of the point on the unit circle. Part 6 The​ y-coordinate of the point determined by −14π3 is −32. The​ x-coordinate of the point determined by −14π3 is −12. Part 7 ​Thus, cot−14π3=33.

1.1 example State whether the functions represented by graphs​ A, B, and C shown in the figure to the right are​ even, odd, or neither. xyABC A coordinate system has a horizontal x-axis and a vertical y-axis. A curve labeled "A" is symmetric about the y-axis. A curve labeled "B" is symmetric about the origin. A curve labeled "C" is symmetric about the y-axis. Question content area bottom Note that the graph is symmetric with respect to the​ y-axis if whenever the point​ (x,y) is on the​ graph, the point ​(−​x,y) is also on the graph​ (that is, the graph is unchanged when reflected about the​ y-axis). The graph is symmetric with respect to the origin if whenever the point​ (x,y) is on the​ graph, the point ​(−​x,−​y) is also on the graph.

The function represented by graph A is symmetric about the​ y-axis. Part 3 The graph of an even function is symmetric about the​ y-axis and the graph of an odd function is symmetric about the origin. Part 4 ​Therefore, the function represented by graph A is even. Part 5 The function represented by graph B is odd as its graph is symmetric about the origin. Part 6 The function represented by graph C is symmetric about the​ y-axis. Part 7 ​Therefore, the function represented by graph C is even.

1.3 example Design a sine function with the given properties. It has a period of 24 with a minimum value of 10 at t=2 and a maximum value of 16 at t=14.

The function shown​ below, when compared to the graph of y=sint​, has an amplitude of A​, a period of 2πB​, a horizontal shift of​ C, and a vertical shift of D. y=Asin(B(θ−C))+D Part 2 To obtain a sine function with the given​ transformations, find values for​ A, B,​ C, and D and substitute into the general form. Use the fact that the period is 24 to find B. 2πB = 24 2π = B•24 Multiply by B. π12 = B Divide by 24 to obtain a value for B. Part 3 Amplitude is the vertical stretch of the sine function. Use the fact that the function has a minimum value of 10 and a maximum value of 16 to find the amplitude. A = max−min2 A = 16−102 A = 3 Part 4 Vertical shift is the vertical distance the function is moved from the origin. It is given by the horizontal line directly in the middle of the​ y-values of the minimum and the​ maximum, which can be found by taking the average of the two values. Use the fact that the function has a minimum value of 10 and a maximum value of 16 to find the vertical shift. D = max+min2 D = 16+102 D = 13 Part 5 Horizontal shift​ (or phase​ shift) is the horizontal distance the function is moved from the origin. For the sine​ function, it is the​ t-value of the vertical line that is directly in the middle of the​ t-values of the maximum and​ minimum, with the minimum on its left and the maximum on its right. Use this information to find C. C = tmin+tmax2 C = 2+142 C = 8 Part 6 Substitute the values of A=3​, B=π12​, C=8​, and D=13 into the general form of the sine function to obtain the final function. y = Asin(B(t−C))+D = 3sinπ12(t−8)+13 = 3sinπt12−2π3+13

1.3 example Identify the amplitude and period of the following function. f(θ)=22sinπθ/22

The function y=AsinBx has an amplitude of |A| and a period of 2πB. First identify the values of A and B in the given function. A=22 ​, B=π22 Part 2 Now determine the amplitude by taking the absolute value of A. 22=22 Part 3 ​Thus, the amplitude of the function is 22. Part 4 To determine the period of the​ function, substitute π22 into 2πB for B and simplify. period = 2πB = 2ππ22 Substitute. Part 5 = 44 Simplify. Part 6 ​Therefore, the period of the function is 44.

1.1 example Determine an appropriate domain of the function. Identify the independent and dependent variables. A stone is thrown vertically upward from the ground at a speed of 75 ​m/s at time t=0. Its distance d​ (in m) above the ground​ (neglecting air​ resistance) is approximated by the function ​f(t)=75t−5t^2.

To find an appropriate​ domain, first notice that the variable t is associated with the domain. Determine the values of t for which the problem makes sense​ (the values of t between the time the stone is thrown ​(t=​0) and the time it strikes the​ ground). Part 2 Solve the equation ​f(t)=75t−5t2=0 to find the times for which the distance the stone is above the ground is zero. 75t−5t2 = 0 ​5t(15−​t) = 0 Factor. Part 3 Set each factor equal to zero and solve. 5t = 0 or 15−t = 0 t = 0 t = 15 Part 4 ​Therefore, the stone leaves the ground at t=0 s and returns to the ground at t=15 s. An appropriate domain that fits the context of this problem is [0,15]. Part 5 Next identify the independent and dependent variables. The independent variable is the variable associated with the​ domain; the dependent variable belongs to the range. Part 6 For the given​ function, t is associated with the​ domain, and is therefore the independent variable. The variable d is associated with the​ range, and is therefore the dependent variable.

1.2 example The graphs of the functions f and g in the figure are obtained by vertical and horizontal shifts and scalings of y=x. Find formulas for f and g. Verify your answers with a graphing utility. -88-88xyy=xy=f(x)y=g(x) A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. A V-shaped curve labeled y = StartAbsoluteValue x EndAbsoluteValue opens upward, has vertex (0, 0), and passes through the points (negative 1, 1) and (1, 1). A V-shaped curve labeled y = f(x) opens upward, has vertex (2, 1), and passes through the points (1, 2) and (3, 2). A V-shaped curve labeled y = g(x) opens downward, has vertex (negative 2, negative 1), and passes through the points (negative 3, negative 2) and (negative 1, negative 2). Question content area bottom Part 1 T

To find an equation for​ f(x), determine the ways in which the function y=x has been shifted and scaled to result in​ f(x). Part 2 The graph of y=​f(x)+d is the graph of y=​f(x) shifted vertically by d units​ (up if d>0 and down if d<​0). The function y=x has been shifted vertically 1 unit up to result in​ f(x), so the value of d is 1. Part 3 The graph of y=​f(x−​b) is the graph of y=​f(x) shifted horizontally by b units​ (right if b>0 and left if b<​0). The function y=x has been shifted horizontally 2 units to the right to result in​ f(x), so the value of b is 2. Part 4 For c>​0, the graph of y=​cf(x) is the graph of y=​f(x) scaled vertically by a factor of c​ (compressed if 0<c<1 and stretched if c>​1). The function y=x has not been scaled vertically to result in​ f(x). Part 5 For c<​0, the graph of y=​cf(x) is the graph of y=​f(x) scaled vertically by a factor of c and reflected across the​ x-axis (compressed if −1<c<0 and stretched if c<−​1). The function y=x has not been reflected across the​ x-axis nor has it been scaled vertically to result in​ f(x). Part 6 For a>​0, the graph of y=​f(ax) is the graph of y=​f(x) scaled horizontally by a factor of a​ (compressed if 0<a<1 and stretched if a>​1). The function y=x has not been scaled horizontally to result in​ f(x). Part 7 For a<0 and a≠−​1, the graph of y=​f(ax) is the graph of y=​f(x) scaled horizontally by a factor of 1a and reflected across the​ y-axis (compressed if 0<1a<1 and stretched if 1a>​1). The function y=x has not been reflected across the​ y-axis nor has it been scaled horizontally to result in​ f(x). Part 8 The graph of y=x has been shifted vertically 1 unit up and shifted horizontally 2 units to the right to result in the function​ f(x). Determine the equation of​ f(x). ​f(x)= x−2+1 Part 9 To find an equation for​ g(x), determine the ways in which the function y=x has been shifted and scaled to result in​ g(x). Part 10 The graph of y=​f(x)+d is the graph of y=​f(x) shifted vertically by d units​ (up if d>0 and down if d<​0). The function y=x has been shifted vertically 1 unit down to result in​ g(x), so the value of d is −1. Part 11 The graph of y=​f(x−​b) is the graph of y=​f(x) shifted horizontally by b units​ (right if b>0 and left if b<​0). The function y=x has been shifted horizontally 2 units to the left to result in​ g(x), so the value of b is −2. Part 12 For c>​0, the graph of y=​cf(x) is the graph of y=​f(x) scaled vertically by a factor of c​ (compressed if 0<c<1 and stretched if c>​1). The function y=x has not been scaled vertically to result in​ g(x). Part 13 For c<​0, the graph of y=​cf(x) is the graph of y=​f(x) scaled vertically by a factor of c and reflected across the​ x-axis (compressed if −1<c<0 and stretched if c<−​1). The function y=x has been reflected across the​ x-axis but has not been scaled vertically to result in​ g(x), so the value of c is −1. Part 14 For a>0 and a≠​1, the graph of y=​f(ax) is the graph of y=​f(x) scaled horizontally by a factor of 1a ​(compressed if 0<1a<1 and stretched if 1a>​1). The function y=x has not been scaled horizontally to result in​ g(x). Part 15 For a<0 and a≠−​1, the graph of y=​f(ax) is the graph of y=​f(x) scaled horizontally by a factor of 1a and reflected across the​ y-axis (compressed if 0<1a<1 and stretched if 1a>​1). The function y=x has not been reflected across the​ y-axis nor has it been scaled horizontally to result in​ g(x). Part 16 The graph of y=x has been shifted horizontally 2 units to the​ left, reflected across the​ x-axis, and shifted vertically 1 unit ​down, to result in the function​ g(x). Determine the equation of​ g(x). ​g(x)=−lx+2l−1

1.2 Consider the function g(x)=x2−3x+4. a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the​ function, such as​ peaks, valleys, and intercepts.

a. Choose the correct graph below. The graphs are displayed using a window size of ​[−10​,10​,1] by ​[−20​,20​,2​]. A. A coordinate system has a horizontal x-axis labeled from negative 15 to 5 in increments of 1 and a vertical y-axis labeled from negative 20 to 20 in increments of 2. A graph falls from left to right at a decreasing rate passing through (negative 2,1), (negative 1,negative 2), and (0,negative 3), before beginning to rise at an increasing rate, passing through (1,negative 2) and (2,1). All points are approximate. B. A coordinate system has a horizontal x-axis labeled from negative 15 to 5 in increments of 1 and a vertical y-axis labeled from negative 20 to 20 in increments of 2. A graph has two branches. The first branch is a smooth curve that rises at a decreasing rate from left to right passing through (negative 14.8,negative 20), (negative 11.2,negative 17), and (negative 7.6,negative 15.2) before beginning to fall at an increasing rate, passing through (negative 6.4,negative 15.8) and (negative 5.2,negative 20) as it approaches the apparent vertical line x equals negative 4. The second branch is a smooth curve that falls at a decreasing rate from left to right passing through (negative 3.5,20), (negative 1.7,0), and (negative 0.4,negative 0.8) before beginning to rise at an increasing rate, passing through (1.7,0) and (negative 5.2,2.4). As x approaches negative 4 from the right, the second branch approaches the apparent vertical line x equals negative 4. All points are approximate. C. A coordinate system has a horizontal x-axis labeled from negative 15 to 5 in increments of 1 and a vertical y-axis labeled from negative 20 to 20 in increments of 2. A graph has two branches. The first branch is a smooth curve that falls at a decreasing rate from left to right passing through (negative 14.8,20), (negative 11.2,17), and (negative 7.6,15.2) before beginning to rise at an increasing rate, passing through (negative 6.4,15.8) and (negative 5.2,20) as it approaches the apparent vertical line x equals negative 4. The second branch is a curve that falls at a decreasing rate from left to right passing through (negative 3.5,20), (negative 2.6,2.9), and (negative 1.7,0), then rises at a decreasing rate passing through (negative 1.1,0.6) and (negative 0.4,0.8), then falls at an increasing rate passing through (0.7,0.5) and (1.7,0), and then finally increases at an increasing rate passing through (3.4,1.1) and (5,5). As x approaches negative 4 from the right, the second branch approaches the apparent vertical line x equals negative 4. All points are approximate. Your answer is correct. D. A coordinate system has a horizontal x-axis labeled from negative 15 to 5 in increments of 1 and a vertical y-axis labeled from negative 20 to 20 in increments of 2. A graph has two connected linear branches. The first branch falls from left to right passing through (negative 6,2), (negative 5,1), and ending at (negative 4,0). The second branch starts at (negative 4,0) and rises from left to right passing through (negative 3,1) and (negative 2,2). Part 2 b. What is the domain of the​ function? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. D=x: x≤enter your response here ​(Type an integer or a​ decimal.) B. D=x: x≠negative 4−4 ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.) Your answer is correct. C. D=x: x≥enter your response here ​(Type an integer or a​ decimal.) D. The domain is all real numbers. Part 3 c. What are the peaks of the​ function? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The function has​ peak(s) at or near x=negative 0.394−0.394. ​(Type an integer or decimal rounded to the nearest thousandth as needed. Use a comma to separate answers as​ needed.) Your answer is not correct. B. The function has no peak. Part 4 What are the valleys of the​ function? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The function has​ valley(s) at or near x=negative 7.606 comma negative 1.732 comma 1.732−7.606,−1.732,1.732. ​(Type an integer or decimal rounded to the nearest thousandth as needed. Use a comma to separate answers as​ needed.) Your answer is not correct. B. The function has no valley. Part 5 What are the intercepts of the​ function? Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice. ​(Type an ordered​ pair, using integers or decimals. Round to the nearest thousandth as needed. Use a comma to separate answers as​ needed.) A. The​ x-intercepts(s) is/are left parenthesis negative 1.732 comma 0 right parenthesis comma left parenthesis 1.732 comma 0 right parenthesis(−1.732,0),(1.732,0) and the​ y-intercept(s) is/are left parenthesis 0 comma 0.75 right parenthesis(0,0.75). B. The​ x-intercept(s) is/are enter your response here and there is no​ y-intercept. C. There is no​ x-intercept and the​ y-intercept(s) is/are enter your response here. D. There is no​ x-intercept and there is no​ y-intercept.

1.2 Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 136 ft3. a. Find and graph the function​ S(x) that gives the surface area of the​ box, for all values of x>0. b. Based on your graph in part​ (a), estimate the value of x that produces the box with a minimum surface area.

a. Determine the function​ S(x). B. S(x)=x2+544x Part 2 Graph the function​ S(x). Choose the correct graph below. The graphs are displayed using a window size of​ [0,25,1] by​ [0,250,10]. D. A coordinate system has a horizontal axis labeled from 0 to 25 in increments of 1 and a vertical axis labeled from 0 to 250 in increments of 10. From left to right, a curve falls steeply at a decreasing rate, passing through (3, 190) to a minimum at (6.5, 126), and then rises at an increasing rate, passing through (12, 189). All coordinates are approximate. Part 3 b. x≈6.60 ​(Round to two decimal places as​ needed.)

1.2 Let​ A(x) be the area of the region bounded by the​ t-axis and the graph of y=​f(t) from t=0 to t=x. Consider the given function and graph. ​f(t)=15 a. Find ​A(1​). b. Find ​A(7​). c. Find a formula for​ A(x), x≥0. 051001020tyy=f(t) A coordinate system has a horizontal t-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. Question content area bottom Part 1 a

a. Notice that the area under the graph of​ f(t) from t=0 to the indicated value of t is defined by a rectangle. Use the formula for the area of a rectangle to find the area. Part 2 The pink shaded region to the right represents the area ​A(1​). The length of the rectangle represented by the shaded region is L=1. 051001020tyy=f(t) A coordinate system has a horizontal t-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. The region above the t-axis, to the right of the y-axis, below the line y = f(t) and to the left of t = 1 is shaded. Part 3 The height of the rectangle is H=15. 051001020tyy=f(t) A coordinate system has a horizontal t-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. The region above the t-axis, to the right of the y-axis, below the line y = f(t) and to the left of t = 1 is shaded. Part 4 Multiply the length and the height to find the area of the rectangle. ​A(1​)=​(1​)(15​)=15 Part 5 b.​ Similarly, find the area under​ f(t) for t=0 to t=x for x=7. The length of the rectangle is L=7. 051001020tyy=f(t) A coordinate system has a horizontal t-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. The region above the t-axis, to the right of the y-axis, below the line y = f(t) and to the left of t = 7 is shaded. Part 6 The height of the rectangle is H=15. 051001020tyy=f(t) A coordinate system has a horizontal t-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. The region above the t-axis, to the right of the y-axis, below the line y = f(t) and to the left of t = 7 is shaded. Part 7 Multiply the length and the height to find the area of the rectangle. ​A(7​)=​(7​)(15​)=105 Part 8 c. To find a formula for​ A(x), determine the general length of the rectangle. L=x 001020tyy=f(t)t=x A coordinate system has a horizontal t-axis with 10 evenly spaced tick marks and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. The region above the t-axis, to the right of the y-axis, below the line y = f(t) and to the left of a point labeled t = x is shaded. Part 9 Notice that the height of the rectangle is the same for all values of x. H=15 001020tyy=f(t)t=x A coordinate system has a horizontal t-axis with 10 evenly spaced tick marks and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 15. The region above the t-axis, to the right of the y-axis, below the line y = f(t) and to the left of a point labeled t = x is shaded. Part 10 ​Therefore, the function for​ A(x) is ​A(x)=15x. The area at any value of​ x, x≥​0, is given

1.2 example Consider the function ​f(x)=x3−3x2+13. a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the​ function, such as​ peaks, valleys, and intercepts.

a. To view all the features of the​ graph, graph it in a​ [−​8,8,2] by ​[−​16,16,2] window. The graph is shown to the right. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,9), (0,13), (1,11), (2,9), and (2.5,9.9). All points are approximate. Part 2 b. Identify the type of function that is being presented. The function ​f(x)=x3−3x2+13 is a polynomial function. Part 3 Recall that polynomial functions are defined for all values of x.​ Thus, the domain is all real numbers. Part 4 c. Use a graphing utility to determine the peaks of the function. Note that a peak is a point that is higher than the other points near it. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,9), (0,13), (1,11), (2,9), and (2.5,9.9). A point is plotted at (0,13). All points are approximate. Part 5 Use a feature of the graphing utility to determine the location of any peaks. The function has one peak at or near x=0. Part 6 Use a graphing utility to determine the valleys of the function. Note that a valley is a point that is lower than the other points near it. A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,9), (0,13), (1,11), (2,9), and (2.5,9.9). A point is plotted at (2,9.0). All points are approximate. Part 7 Use a feature of the graphing utility to determine the location of any valleys. The function has one valley at or near x=2. Part 8 Find intercepts of the function. Note that an intercept is a point where the graph intersects an axis. Notice in the graph that there is one​ x-intercept and one​ y-intercept. Begin by finding the​ x-intercept using a graphing utility. The​ x-coordinate has been rounded to the nearest hundredth. The​ x-intercept is approximately (−1.67,0). A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,9), (0,13), (1,11), (2,9), and (2.5,9.9). All points are approximate. Part 9 Find the​ y-intercept. The​ y-intercept is (0,13). A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A smooth curve rises at a decreasing rate to a local maximum, then begins to fall toward a local minimum, then begins to rise at an increasing rate, passing through the points (negative 1,9), (0,13), (1,11), (2,9), and (2.5,9.9). All points are approximate. Part 10 The graph of ​f(x)=x3−3x2+13 is shown to the right in a ​[−​8,8,2] by ​[−​16,16,2] window. The domain of the function is all real numbers. It has a peak that occurs at or near x=0 and a valley that occurs at or near x=2. The​ x-intercept of f is approximately ​(−1.67​,0) and the​ y-intercept is ​(0,13​).

1.2 Let​ A(x) be the area of the region bounded by the​ t-axis and the graph of y=​f(t) from t=0 to t=x. Consider the given function and graph. ​f(t)=7 a. Find ​A(4​). b. Find ​A(7​). c. Find a formula for​ A(x), x≥0. 051001020tyy=f(t) A coordinate system has a horizontal t-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. A horizontal line labeled y = f(t) crosses the y-axis at 7. Question content area bottom Part 1 a.

a.​ A(4​)=2828 ​(Simplify your​ answer.) Part 2 b.​ A(7​)=4949 ​(Simplify your​ answer.) Part 3 c. Find a formula for​ A(x), x≥0. ​A(x)=7 x

1.2 Use shifts and scalings to graph the given function. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. f(x)=(x+3)^2+6 Question content area bottom Part 1 What is the original​ function? y=enter your response here ​(Type an expression. Simplify your​ answer.) Describe how to obtain the graph of f(x)=(x+3)2+6 from the original function. Select all that apply. A. Shift left 3 units B. Shift up 6 units C. Shift right 3 units D. Shift down 6 unit Choose the correct graph of​ f(x) below. A. -1010-416xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 4 to 16 in increments of 1. A U-shaped graph opens upward and passes through (negative 5,10), (negative 4,7), (negative 3,6), (negative 2,7), and (negative 1,10). B. -1010-164xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 16 to 4 in increments of 1. A U-shaped graph opens upward and passes through (1,10), (2,7), (3,6), (4,7), and (5,10). C. -1010-416xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 4 to 20 in increments of 1. A U-shaped graph opens upward and passes through (negative 5,1), (negative 4,negative 2), (negative 3,negative 3), (negative 2,negative 2), and (negative 1,1). D. -1010-164xy

y=x^2 A. Shift left 3 units Your answer is correct. B. Shift up 6 unit A. -1010-416xy A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 4 to 16 in increments of 1. A U-shaped graph opens upward and passes through (negative 5,10), (negative 4,7), (negative 3,6), (negative 2,7), and (negative 1,10).

1.2 The factorial function is defined for positive integers as n!=n(n−1)(n−2)⋯3•2•1. ​(a) Make a table of the factorial function for n=​1, ​2, 3,​ 4, 5. ​(b) Graph these data points and then connect them with a smooth curve. ​(c) What is the least value of n for which n!>106​?

​(a) Complete the table below. n 1 2 3 4 5 ​f(n)=​n! 1 2 6 24 120 Part 2 ​(b) Choose the correct graph below. B. 060140nf(n) A coordinate system has a horizontal n-axis labeled from 0 to 6 in increments of 1 and a vertical f(n)-axis labeled from 0 to 140 in increments of 20. A curve rises from left to right, starting at the plotted point (1, 1) and passing through the plotted points (2, 2), (3, 6), (4, 24), and (5, 120). Part 3 ​(c) The least value of n for which n!>106 is 10 Note: Evaluate​ n! for integers greater than 5. Find the least value of n for which n!>106.