Honors Precalculus (Caron) Chapter 4 Notes

slope formula

(y₂- y₁) / (x₂- x₁)

Suppose the height of an object shot straight up is given by h = 512t - 16t². (Here h is in feet and t is in seconds) Find the maximum height and time at which the object hits the ground

-512/32 = 16 (512)(16) - 16(16)² max height = 4096 ft If a.o.s. is 16, the object hits the ground at 2(16), or after 32 seconds

State the formulas for the following: a) area of circle b) circumference of circle c) area of rectangle d) perimeter of rectangle e) area of triangle f) perimeter of triangle

. A = πr² . C = 2πr . A = l*w . P = 2l + 2w . A = 1/2b*h . P = a + b + c

How would you graph the quadratic function y = x² - 2x + 3 by completing the square and translating it?

. Add a space y = x² - 2x ____ + 3 . complete the square: (-2/2)² = 1 y = (x² - 2x + 1) + 3 add the value to both sides but as a negative to the outer value (right side) y = (x - 1)² + 3 - 1, or y = (x - 1)² + 2 . graph shift the origin from (0,0) to the point (1,2)

How can you use your graphing calculator to determine a quadratic function that best fits the data set?

. STAT menu . CALC . QuadReg

What is the polynomial end behavior for the arms of the graph when the degree of n is: . even . odd

. both up or both down . one arm of the graph is up and one arm is down

What are four types of power functions and what are qualities of them given: . n = 0 . n = 1 . n = 2 . n = 3

. constant . linear . quadratic . cubic no corners, breaks, or holes

What do all power functions share?

. origin . some kind of (1,1) point

Describe the graph y = ax²: . shape . reflection . width

. parabola with vertex at origin that is similar in shape to y = x² . opens upward if a > 0, down if a < 0 . narrower than y = x² if |a| > 1, wider than y = x² if |a| < 1

Given the function y = -t⁴ + 3t², where the maximum value as seen on the graph occurs between 1.0 and 1.5 and -1.0 and -1.5, determine the maximums exactly and obtain calculator approximations.

. reduce the question to a quadratic: assume t² = x, so t⁴ = x² y = -x² + 3x . complete the square y = -(x - 3/2) + 9/4 . maximum when x = 3/2, so: t² = x = 3/2 t = ±√3/2, so ±√6/2 and ±1.225

What does the graph of y = 1/xⁿ resemble when n is: . an even integer greater than 2 . an odd integer greater than 3

. the graph of y = 1/x² . the graph of y = 1/x³

What is the polynomial end behavior for the arms of the graph when the leading coefficient aₙ is: . positive . negative

. the right arm of the graph is up . the right arm of the graph is down

If f(x) = ³√x² - x + 1, which x-value yields the minimum value for the function f? what is the minimum value?

. the x-value that minimizes it will be the same x-value that minimizes y = x² - x + 1 . complete the square (x² - x + 1/4) + 1 - 1/4 (x - 1/2) + 3/4 . vertex (1/2, 3/4); upward so x = 1/2 produces minimum for both . minimum value: f(1/2) = ³√(1/2)² - (1/2) + 1 = 0.91

Graph the following graph based on y = 1/x³: y = 4(x-2)³

. translate origin right 2; vertical asymptote at x = 2 . shift points 4 affects y and add 2 to x: (1,1) becomes (3,4) (-1,-1) becomes (1, -4)

Describe how to graph the following: . y = 3(x - 1)² . y = -3(x - 1)²

. translate parabola one unit to the right, narrower than y = x², y-int (0,3) . same as above but reflected across x axis

Translate the reflection of the power function y = -2(x - 3)⁴ and describe the graph.

. use y = -2x⁴ as a reference . move three units to the right . (-1,2) and (4,-2) . y-int (4, -32) resembles y = x² graph

State what graph the following functions would resemble: . y = x⁶ . y = x⁵ . y = x⁴

. y = x² . y = x³ . y = x²

Express the area of a right triangle as a function of one variable given the top point P(x, y) and the height y = 16 - x².

1) A = 1/2b*h 2) A = 1/2x * (16 - x²) A = 8x - 1/2x³ D: given 16 - x² > 0, (0,4) because -4 not applicable

A rectangle is inscribed in a circle in which domain = 8 cm. Express the perimeter of the rectangle as a function of its width.

1) a right triangle with a hypotenuse of 8 can be derived from this Pythagorean theorem: a² + b² = c² P = 2l + 2w D = 8, R = 4 2) formula for perimeter in terms of x; rename length using Pythagorean Theorem 8² = w² + l² l = √64 - w² 4) P = 2l + 2w so P = 2(√64 - w²) + 2w Domain: 64 - x² > 0; 64 > x² x < 8, x > -8 x < 8 bc can't be negative

What are the four steps for setting up equations that define functions?

1) draw a picture that represents it 2) state in your own words what the problem is asking for (assign a variable to denote a key quantity if necessary) 3) label any other quantities in your figure that appear relevant (find equations) 4) find an equation involving the key variable that you identified in step 2 and substitute in the auxiliary equations to get an equation with only the required variables

Expressing a distance as a function of one variable: let P(x) be a point on the curve y = √x and express the distance from the point (1,o) as a function of one variable.

1) graph y = √x, two pts (1,0) (y,x) Distance equation: D = √(x₂-x₁)² + (y₂ - y₁)² D = √(x - 1)² + (y - 0)² D = √x² - 2x + 1 + y(√x)² D = √x² - x + 1

Find the area of a rectangle as a function of its width and the domain given that: . perimeter = 100 cm . area = length * width . perimeter = 2l + 2w

1) label the perimeter, l, and w (width) 2) look for area in terms of w 3) P = 2l + 2w; rearrange for l l = 50 - w 4) A = lw A = x(50 - x) x(50 - x) ≥ 0; x ≥ 0 domain = (0,50) bc can't be negative

Graph the function f(x) = -2x² + 4x + 6 and specify the vertex, axis of symmetry, maximum or minimum value of f, and x and y intercepts.

1. complete the square y = -2(x² - 2x ) + 6 add 0 = (-2)(1) to the right side; 6 + 2 y = -2(x - 1)² + 8 vertex: (1, 8) a.o.s.: x = 1 max. value of f: 8 y-int: 6 x-ints: -1, 3

Compare the graphs y = x², y = 2x², and y = 1/2x²

1/2x² < x² < 2x² the graph of 1/2x² is below y =2x², so: y = 2x² is the narrowest y = x² is middling y = 1/2x² is the widest

A cylindrical can holds 20π inches. the material for the top and bottom costs $10/m² and material for the side costs $8/m². Find the radius r and height h of the most economical can

20π = πr²h h = 20/r² SA = 20πr² + 2πrh C = 10(2πr²) + 8(2πrh) C = 20πr² + 16πrh C = 20πr² + 16πr(20/r²) C = 20πr² + 320π/r radius = 2; plugged into formula, min. cost = $754 h = 20/(2)² h = 5

Solve the following: A farmer has 2400 feet of fencing and wants to fence off a rectangular field that borders a river on one side. What are the dimensions that will maximize the area?

2400 = 2w + l A = wl A = -2w² + 2400 -2400/-4 = 600 2400 = 2(600) + l; l = 1200 dimensions to maximize area = 600 x 1200 ft

Expressing the area of a circle as a function of the circumference: A piece of wire x inches long is bent into a circle. Express the area of the circle in terms of x.

C = 2πr A = πr² 2πr = x 1/r = 2π/x r = x/2π A = π(x/2π)² A = x²/4π D: x > 0 so (0, ∞)

Find the dimensions and maximum area of a rectangle if its perimeter is 60 inches.

P = 2l + 2w 30 = l + w l = 30 - w A = l * w A = (30 - w)(w) A = 30w - w² -b/2a; -30/-2 = 15 A = -225 + 450; A = 225 vertex = (15, 225) 15 x 15 inches 225 in²

Solve the following: Ms. Pinto has 400 yards of fence to make a rectangular corral. If a river will be used as one side of the corral, what would be the length and width for the maximum area?

P = 2w + 1 400 = 2w + l l = 400 - 2w A = l * w A = (400 - 2w)(w) A = 400w - 2w² -b/2a; vertex = (100, 20,000) max. area = 20,000 yd²

Given the function P = -1/3x + 40, where 0 ≤ x ≤ 17, the y-int is 40, and the x-int is 117, what is the revenue function expressed in terms of one variable?

R = x(-1/3x + 10) R = -1/3x² + 40x

Suppose that the following demand function relates the selling price, p, of an item to the quantity solid, x: p = -1/3x + 40 (p in $) For which value of x will the revenue be at its maximum? Computer the corresponding unit price and maximum revenue.

R = xp P = -1/3x + 40 R = x(-1/3x + 40) R = -1/3x² + 40 -b/2a = 60; sold 60 units plug 60 in; unit price = $20 (60)(20) = 1200; revenue

An open rectangular box with a square base is to be made from 48 ft² of material. What dimensions will result in a box with the largest possible volume?

SA = x² + 4xh V = x²h h = 48-x²/4x V = x²(48-x²/4x); V = 4x - x³/4 Plug into graphing calculator; x = 4 so 48 = 16 + 4(4)h h = 2 dimensions = 4 * 4 * 2

polynomial function

a function defined by an equation of the form f(x) = aₙˣʰ + aₙ−₁ˣʰ⁻¹ + . . . + a₁x + a₀ when n is a nonnegative interger and the a₁' are constants. ex. 2x³ - 3x² + x - 4; 3u⁷ - √3u² - π; 5x + 1

linear function

a function defined by an equation of the form f(x) = Ax + B in which A and B are constants

linear/straight-line depreciation

a function in economics where you assumed that the value V(t) of an asset decreases linearly over time t: V(t) = mt + b where t = 0 corresponds to the time when the asset is new and the slope m is negative

power function

a function of the form y = xⁿ, where n is any real number constant

What exists when there is no horizontal asymptote?

a slant or oblique asymptote

Solve the following: A new machine costs $8000. After 10 years, it has a salvage value of $500 (given pts. (0,8000) and (10,500) a) assuming linear depreciation, find a formula for the value V(t) of the machine after t year, where 0 ≤ t ≤ 10 b) Use the depreciation function determined in part a to find the value of the machine after 7 years

a) (10, 500) and (0, 8000); (8000- 500)/(0-10) = -750 V(t) = -750t + 8000 b) V(7) = -750(7) + 8000 = $2750

If the cost C(x) in dollars of producing x bicycles is given by the linear equation C(x) = 625 + 45x: a) find the cost of producing 10 bicycles b) find the marginal cost c) use a and b to find the cost of producing 11 bicycles and check by evaluating C(11)

a) C(10) = 625 + 45(10) = $1075 b) linear function so marginal cost = slope; marginal cost = $45 per bicycle c) cost of 11 bikes = cost of 10 bikes + marginal cost $1075 + $45 = $1120; C(11) = $1120

What are maximum/minimum problems described by and how are they solved?

described by quadratic functions and solved by finding the vertex of functions

distance formula

distance = rate * time (d = rt, linear function where t is a constant)

What is the horizontal asymptote if both polynomials are the same degree?

divide by coefficients of the highest degree term ex. (6x²-3x+4)/(2x²-8) 6/2 = 3

Solve the following: Which point on the curve y = √x is closest to the point (1,0)?

find the minimum (1,0) (y,√x) d = √(x₂-x₁)² + (y₂-y₁)² d = √(x-1)² + √x-0)² d = √x² - x + 1 1/2; plug back in so √2/2

Find the horizontal asymptote using long division: y = (4x -2)/(x - 1)

find the remainder and solution divide the remainder by the divider and add the solution y = 2/(x -1) + 4 y = 4

quadratic function

functions defined by the equations of the form f(x) = ax² + bx + c where a ≠ 0 and a, b, and c are constants

What can you do once you know the graph of y = k/x?

graph any rational function of the form y = (ax + b)/(cx + d)

absolute maximum

highest total point

demand function

hypothetical function relating the selling price of a certain item to the number of units sold

What is the difference in the following graphs obtained from y = 2/x: . y = 2(x - 1) . y = -2(x - 1)

identical except the first one is in quadrants I and III and the second equation is reflected into quadrants II and IV

In a table of x-y values where the x values are equally spaced, when can the data be generated by a linear function?

if and only if the first differences of the y-values are constant ex. 6 6 6

In a table of x-y values where the x values are equally spaced, when can the data be generated by a quadratic function?

if and only if the second differences of the y-values are constant

cost function

in economics, a function that gives the cost C(x) for producing x units of a commodity

What happens if the denominator cancels with the numerator?

it is a hole instead of an asymptote

What kind of function is it if the second differences are 3, 1, 3?

it is not a linear or quadratic function

asymptote

line that the graph of the function approaches but never touches

regression line/least squares line

linear function that best fits the data of a scatter plot

absolute minimum

lowest total point

When should you find a maximum and when should you find a minimum?

max: graph reflects downward min: graph points upward

How can you find the number of extrema?

n - 1

What is the degree of the polynomial function if aₙ ≠ 0?

n, the largest exponent on the input value

first differences

new list formed by subtracting adjacent numbers of an original list as follows: given 2, 7, 8, 4, 15, first diff. = 5, 1, -4, 11

When does a parabola open up and when does it open down?

opens up when a > 0 opens down when a < 0

Solve the following: Suppose that f is a linear function. If f(1) = 0 and f(2) = 3, find an equation defining f.

ordered pairs: (1,0) and (2,3) slope: (3-0)/(2-1) = 3 pt. slope: y - 0 = 3(x - 1) y = 3x - 3

scatter diagram/scatter plot

plot of (x,y) pairs given in a table

How can you find the sum and product of roots?

product = c/a sum = -b/a

Graph the following graph based on y = 1/x²: y = -1/(x+3)²

range (0, ∞) for original vertical asymptote at x = -3 reflected over x axis so range changes

slope

rate of change

velocity

rate of change of distance with respect to time

What are the next simplest functions after polynomial functions?

rational functions

How can you find the vertical asymptote?

setting the denominator equal to zero and solve ex. (3x-1)/(x+5); x = -5 (3x²+6x+5)/(x²-3x+2); x = 2, 1 (x+6)/x(x+5); x = 0, 5

How do you find the y-int for a quadratic function in vertex form?

substitute in x = 0

How do you find the x intercepts of a quadratic equation in vertex form?

substitute y = 0

second differences

subtracting adjacent numbers in a list of first differences ex. first diff = 5, 1, -4, 11 second diff = -4, -5, 15

extrema

sum of maximums and minimums (number of turns)

marginal cost

the additional cost to produce one more unit

What does the graph of y = k/x resemble if k is a positive constant?

the graph of y = 1/x x stays the same, y changes

What happens when n is an integer greater than 3 and what does it determine?

the graph of y = xh resembles the graph of y = x² or y = x³ depending on whether n is even or odd; shape

What is the horizontal asymptote if the numerator is a lower degree than the denominator?

y = 0

How would you graph the translate of y = 1/x y = 1/(x - 1)?

y = a(1/(x-h)) + k shift vertical asymptote right 1 unit

What form are rational functions in?

y = f(x)/g(x) ex. reciprocal 1/x, 2 vertical asymptotes y = (5x-1)/(x²-4), denominator ≠ 0 y = 6/(1 + x²)

How would you find the graph of the following equation in factored form: f(x) = x(x + 1)(x - 3)

zeroes 0, -1, 3 use end behavior; in this case, down then up find points between zeroes and plug them into the equation to get a better idea of the shape of the graph

percentage error formula

|(actual value) - (predicted value)|/actual value * 100%

Which point on the curve y = √x -2 + 1 is closest to the point (4,1)? What is this minimum distance?

d = √(x₂-x₁)² + (y₁-y₂)² d = √(x-4)² + (√x-2 + 1 - 1)² √x² -7x +14 -b/2a = 7/2 plug into distance equation, = √7/2 √(7/2 - 2) + 1 √6/2 + 1 = 2√6/2

Express the surface area of a cylinder of given volume as a function of the radius of the base given that V = 10 cm³

V = l * w * h V = b * h V = πr² SA = 2πr²+ 2πrh 10 = πr²h h = 10/πr² 2πr² + 2πr(10/πr²) SA = 2πr² + 20/r D: (0,∞)

relative minimum

bottom of the valleys, lowest points in neighborhoods relative to the points around them

parabola

curve found in the graph of any quadratic function similar in shape to the basic y = x² graph

There are two train routes: one follows the curve y = x⁴ + 10 and the other follows y = 4x². Assume that distance along both axes is measured in miles. The railroad wants to construct a maintenance route north-south between the two. Where should it be located to maximize length?

d = √(x-x)² + (x⁴ + 10 - 4x²) d = √(x⁴ - 4x² + 10)² d - x⁴ - 4x² + 10 4/2 = 2; if t = x², then x = √2 y-coordinate for p: 14 y-coordinate for q:8 14-8 = 6 miles

How do you find the horizontal asymptote of a rational function?

the method changes depending on how degrees of polynomials in the numerator and denominator compare

What does the highest exponent indicate?

the number of zeroes in a polynomial

What equation can be used to express revenue as a function of one variable in economics?

the revenue R generated by selling x units at price of $ p per unit is given by the equation R = x + p p = price per unit, x = # of units

What is the marginal cost equal to when the cost function is linear?

the slope of the corresponding line

What is the slope in the distance formula?

the velocity

Vertex Form Theorem

the x-coordinate of the vertex of the parabola y = ax² + bx + c (a ≠ 0) is given by x = -b/2a ex. 2x² - 12x + 5 = 3

What is the horizontal asymptote if the numerator is a higher degree than the denominator?

there is no horizontal asymptote

relative maximum

tops of mountains; highest points in neighborhoods relative to the points around them ex. f has a relative max of 2 at x = 3

How would you sketch/translate the graph of the power function y = (x + 2)⁵?

translate two units to the left; pts (-1,1) and (3, -1); y-int 32, found by substituting 0 in for x resembles y = x³ graph

vertex

turning point on the parabola form (x,y)

If the equation of the parabola y = ax² + bx + c is rewritten in the form y = a(x - h)² + k, what is the vertex, a.o.s., and reflection?

vertex: (h, k) a.o.s.: (x = h) upward if a > 0, down if a < 0

Graph the following: y = (4x - 2)/(x - 1)

vertical asymptote: x - 1 ≠ 0, so x = 1 y-intercept: plug 0 in for x, (0,2) x-intercept: numerator 4x - 2 = 0, y = 1/2 horizontal asymptote: long division or synthetic division = 4, R2; 2/x-1 + 4 so y = 4

axis of symmetry

vertical line passing through the vertex of the parabola form x = __

Express a certain product as a function of a single variable: Two numbers add up to 8. Express the product P of these two numbers in terms of a single variable.

x + y = 8 P = xy x = 8 - y P = y(8-y) P = 8y - y²

Solve the following: Two numbers add to 9. What is the largest possible value for this product?

x + y = 9 P = xy P = y(9 - y) P = -y² + 9y -9/-2 = 9/2 (x-value at maximum but y-value of equation) Plug into first equation: x + 4.5 = 9; x = 4.5 (4.5, 20.25) in which 20.25 is the largest value

What does it mean to have x values that are equally spaced?

x = 1, 2, 3 or x = 5, 10, 15 NOT x = 1, 2, 4