Trimester 1
Find the point of intersection for the pair of lines, 2x+ 4y = -10 3x5y = 18
(1, -3)
Find the point of intersection for the pair of lines, y = 5x-8 y = -2x - 1
(1, -3)
Using P = f (t), where P represents the population, in thousands, of birds on an island and t is the number of years since 2007: (a) What does f(4) represent? (b) What does f−1(4) represent?
(a) The expression f(4) is the bird population(in thousands)in the year 2011. (b) Since f −1 is the inverse function, f −1 is a function which takes population as input and returns time as output. Therefore, f−1(4) is the number of years after 2007 at which there were 4,000 birds on the island.
What is the vertex of a parabola?
(h,k)
what would the perpendicular line of m1 be?
-1/m2
Given h(x) = 2x-7, what is the x valye for the output of -4?
3/2
A 1-1 function must pass which tests?
Both the horizontal and vertical line tests
How do you find the inverse of a function?
Switch variables and solve
What is a function?
System-defined formula
State the domain and range of g, where g(x)= 1/x
The domain is all real numbers except those which do not yield an output value. The expression 1/x is defined for any real number x except 0 (division by 0 is undefined).
Given f(x) = 2x^2 a) find the average rate of change of f(x) from x = -3 to x = 2 b) find the average rate of change of f(x) from x = -5 to x = 5
a) -2 b) 0
Given f(x) = -2x^2 + 1, g(x) = -3x -2, and h(x) = 2x+3. Evaluate each of the following a) f(g(4)) b) g(h(-2)) c) h(g(f(-2)))
a) -391 b) 1 c) 41
How do you find h when completing the square?
h = -b/2a
What is the input?
n or the variable in the parenthesis
Given g(x) = {4, x<=2, x^3-5, x>=2} a) g(1) b) g(2) c) g(3)
a) 4 b) 3 c) 22
A grapefruit is thrown into the air. Its velocity, v, is a linear function of t, the time since it was thrown. A positive velocity indicates the grapefruit is rising and a negative velocity indicates it is falling. Check that the data in Table 1.28 corresponds to a linear function. t, time(sec) = 1,2,3,4 v, velocity (ft/sec) = 48, 16, -16, -48
A formula for the velocity is v = 80 − 32t.
Sales of digital video disc (DVD) players have been increasing since they were introduced in early 1998. To measure how fast sales were increasing, we calculate a rate of change, in what way?
Change in sales ------------------ Change in time
How does one figure out if a function is concave up or down based on a table?
Concave down if the average rates of changes are decreasing Concave up if the average rates of changes are increasing
What is the domain and range for f(x) = x^2 + 3?
Domain: all real numbers, (-infinity, +infinity) Range: y >= 3, [3, +infinity)
If a function is concave up, then does it must be increasing?
False
If f(3) = 5 and f is invertible, then does f-1(5) = 1/3?
False
Is a straight line concave up?
False
What is linear regression?
Fitting the best line to a set of data
What does it mean if a function is invertible?
If the inverse of a function is also a function
What is the vertical line test?
To tell when a graph is a function of y if the same x value occurs twice with one y value
given f(x) = (3x-7)/6 and g(x) = (6x+7)/3 are f(x) and g(x) inverse functions?
Yes
What two things must you look out for in domain restrictions?
Zero in the denominator, negative even in a radical
Give h(x), find f and g such that h(x) = f(g(x)) a) h(x) = -3(x-2)^7 + 5 b) h(x) = 2(3^(4x-5)) + 7
a) b)
Given f(x) = (3x-5)/4 and g(x) = (4x+ 4)/3 a) evaluate f(g(x)) b) evaluate g(f(x)) c) Are f and g inverse functions?
a) (4x-1)/4 b) (3x-1)/3 c) No because f(g(x)) is not equal to g(f(x)) is not equal to x
Given f(x) = 1/x - 2x^2 find a) f(2) b) f(a) c) f(x + 1)
a) -7.5 b) 1/a - 2a^2 c) 1/(x+1) - 2x^2 - 4x - 2
Given f(x) = x^2-5 and g(x) = -x + 2 a) find f(g(6)) b) find g(g(f(x))) c) True or false f(g(3)) > g(f(2))
a) 11 b) x^2-5 c) true -4<3
A sunflower plant is measured every day t, for t ≥ 0. The height, h(t) centimeters, of the plant4 can be modeled by using the logistic function h(t) = 260 . 1 + 24(0.9)t (a) Using a graphing calculator or computer, graph the height over 80 days. (b) What is the domain of this function?What is the range?What does this tell you about the height of the sunflower?
a) 250 cm b) The domain of this function is t ≥ 0. If we consider the fact that the sunflower dies at some point, then there is an upper bound on the domain, 0 ≤ t ≤ T , where T is the day on which the sunflower dies. To find the range, notice that the smallest value of h occurs at t = 0. Evaluating gives h(0) = 10.4 cm. This means that the plant was 10.4 cm high when it was first measured on day t = 0. Tracing along the graph, h(t) increases. As t-values get large, h(t)-values approach, but never reach, 260. This suggests that the range is 10.4 ≤ h(t) < 260. This information tells us that sunflowers typically grow to a height of about 260 cm.
We have $24 to spend on soda and chips for a party. A six-pack of soda costs $3 and a bag of chips costs $2. The number of six-packs we can afford, y, is a function of the number of bags of chips we decide to buy, x. (a) Find an equation relating x and y. (b) Graph the equation. Interpret the intercepts and the slope in the context of the party.
a) 2x + 3y = 24 b This means that if we buy 6 bags of chips, we can afford 4 six-packs of so
The cost in dollars of renting a car for a day from three different rental agencies and driving it d miles is given by the following functions: C1 =50+0.10d, C2 =30+0.20d, C3 =0.50d. (a) Describe in words the daily rental arrangements made by each of these three agencies.
a) Agency 1 charges $50 plus $0.10 per mile driven. Agency 2 charges $30 plus $0.20 per mile. Agency 3 charges $0.50 per mile driven.
The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft^2. We write n = f(A). (a) Find a formula for f. (b) Explain in words what the statement f (10,000) = 40 tells us about painting houses.
a) If A = 5000 ft2, then n = 5000/250 = 20 gallons of paint. In general, n and A are related by the formula b) The input of the function n = f(A) is an area and the output is an amount of paint. The statement f (10,000) = 40 tells us that an area of A = 10,000 ft2 requires n = 40 gallons of paint.
A town of 30,000 people grows by 2000 people every year. Since the population, P , is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (a) What is the average rate of change of P over every time interval? (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. (c) Find a formula for P as a function of t.
a) The average rate of change of population with respect to time is 2000 people per year. (b) The initial population in year t = 0 is P = 30,000 people. c) so a formula for P in terms of t is P = 30,000 + 2000t.
Suppose h = f(t) represents the height in inches of a child at time t years after 1990 a) Explain the meaning of the statement f(5) = 40 b) What does f-1(50) mean in practical terms? c) Write the following statement in function notation, "in 1998, the child's height is 44 inches"
a) The height at 5 years after 1990 is 40 inches b) 50 inches at x year after 1990 c) f(8) = 44
With time, t, in years, the populations of four towns, PA , PB , PC and PD , are given by the following formulas: PA = 20,000 + 1600t, PB = 50,000 − 300t, PC = 650t + 45,000, PD = 15,000(1.07)t. (a) Which populations are represented by linear functions? (b) Describe in words what each linear model tells you about that town's population. Which town starts out with the most people? Which town is growing fastest?
a) The populations of towns A, B, and C are represented by linear functions because they are written in the form P = b + mt. Town D's population does not grow linearly since its formula, PD = 15,000(1.07)t, cannot be expressed in the form PD = b + mt. b) This means that in year t = 0, town A has 20,000 people. It grows by 1600 people per year. This means that town B starts with 50,000 people. The negative slope indicates that the population is decreasing at the rate of 300 people per year.
A floor refinishing company charges $1.83 per square foot to strip and refinish a tile floor for up to 1000 square feet. There is an additional charge of $350 for toxic waste disposal if a job includes more than 150 square feet of tile. a) Express the cost, C = f(x), of refinishing a floor as a function of the number of square feet, x, to be refinished. b) Evaluate f(50) and f(500) and interpret their meanings in the context of the problem c) Give the domain and the range of this function in the context of the problem.
a) c(x) = { 1.83x 0< x <= 150 1.83x + 350 150< x<= 1000 b) f(50) = 91.5, to refinish 50 ft^2 of floor, it will cost $91.50 f(500) = 1265 to refinish 500 ft^2 of floor, it will cost $1265 c) domain: 0 < x <= 1000 range: 0 < y <= 274.5 and 624.5 < y < = 2180
A power company offers residents a certain area the following options for supplying electric power, Option 1: A flat fee of $200 a month regardless of how much power you use Option 2: A flat fee of $60 a month plus $0.1 per kwh Option 3: $0.2 per kwh with no flat fee imposed a) Write a formula for each option letting C represent the cost per month and h represent the number of kwh of usage per month b) Sketch the graphs of the three options accurately c) find the points of intersection of each pair of lines algebraically d) When (exactly) would option 1 be the best option? e) When (exactly) would option 3 be the best option?
a) c1= 200, c2 = 60 +0.1x, c3 = 0.2x b) check paper c) c1 & c2 = (1400,200), c2 & c3 = (600, 120), c1 & c3 = (1000, 200) d) When you use more than 1400 kwh e) When you use less than 600 kwh
If f and g are inverse functions, and f(2) = -4 and g(3) = 1, evaluate each of the following if possible a) f(-4) b) g (-4) c) g-1(1) d) f(g(-3)) e) g(f(g(3)))
a) cannot be determined b) 2 c) 3 d) -3 e) 1
State the domain and range of each of the following functions a) f(x) = -2√(8-2x) + 4 b) g(x) = 1/(x+5) - 4
a) domain: x <= 4 range: y <= 4 b) domain: x is not equal to -5 range: y is not equal to -4
The cost of producing q items is given by the function c = f(q) = 400 + 3q a) Evaluate f(1000) and interpret its means in the context of the problem. (be sure to include units in your answers) b) Find a formula for the inverse function c) Evaluate f-1(1000) and interpret its means in the context of the problem (be sure to include units in your answers)
a) f(1000) = 400 +3(1000) = 3400, if you want to produce 100 items, it will cost $3400 b) q = f-1(c) = (c-400)/3 c) q = f-1(1000) = (1000-400)/3 = 200, if you have $1000 you can produce 200 items
A car company has found that there is a linear relationship between the amount of money it spends on advertising and the number of cars it spends on advertising and the number of cars it sells. When it spends 50 thousand dollars on advertising, it sold 500 cars. Moreover, for each additional thousand dollars spent, they sell 20 more cars. a) Let x be the amount of money they spend on advertising, in thousand of dollars. Find the formula for q = f(x), the number of cars sold. b) What is the slope of your equation? What is the meaning of this in terms of the problem? c) What is the y-intercept of your equation? What is the meaning of this in terms of the problem? d) Evaluate f(225) and interpret its meaning e) Evaluate f-1(1000) and interpret its meaning
a) f(x) = 4x + 300 b) m = 4, for every 1000 dollars spent on advertising, you should expect to sell four more cars c) b = 300, if you spend no money on advertising you should expect to sell 300 more cars d) If you spend 225,000 dollars in advertising you should expect to sell 1200 cars e) If you want to sell 1000 cars, you will need to spend 175,000 dollars on advertising
Find a formula, g-1(x), for the inverse of each of the following functions a) g(x) = -3/(5x+1) + 2 b) g(x) = -2x^3 + 4
a) g-1(x) = y = -3/5(x-2) - 1/5 b) g-1(x) = y = ^3√((x-4)/-2)
Let w(m) give weight (in pounds) of an average-sized baby girl who is m months old a) An average six-month girl weighs 15 pounds. Express this fact using function notation b) What does the statement w(12) = 21 mean in the context of this problem? c) What does the statement w-1(10) = p mean int he context of this problem? d) A four-month old baby girl named Emily weighs w(6) pounds. Is Emily of average weight, above average weight, or below average weight? Explain this relationship
a) w(6) = 15 b) An average 12 month old baby weighs 21 pounds c) p is the age (in months) of an average 10 pound baby d) overweight- this says a four month old baby weighs what an average 6 month old baby weighs
A company's profit (in millions of dollars per year) as a function of time (in years) is given by the formula p(t) = -t^2 + 7t - 2 with t= 0 in 1980 a) Accurately sketch the graph of p(t). Be sure to label all intercepts, find the vertex, axis of symmetry, and one more point on the graph b) Evaluate p(3). Show where this is represented on your graph and interpret your answer in the context of this problem. c) Estimate p(t) = 4. Show how you found this from your graph and interpret your answer in the context of this problem d) Give the domain and range of this function
a) y-int: (0,2) x-int : ((-7+-√41)/-2), 0) vertex: (3.5, 10.25) b) p(3) = 10 In 1983, the company's profits are 10 million dollars c) t = 1, 6, when the companies' profits are 4 million dollars, the years were 1981 and 1986 d) Domain: all real numbers range: all real numbers less than or equal to 10.25
What is the output?
f or the variable outside of the parenthesis
what kind of slopes do perpendicular lines have?
opposite reciprocal slopes
How do you find k when completing the square?
plug in the x value and solve for y
Put each quadratic function into vertex form by completing the square and then graph it. (a) s(x)=x^2 −6x+8
s(x) = (x−3)^2 −1 The vertex of s is (3, −1)
constant rate of change
the rate of change in a linear relationship
what kind of slopes do parallel lines have?
the same
x-intercept
the x-coordinate of a point where a graph crosses the x-axis
y-intercept
the y-coordinate of a point where a graph crosses the y-axis
Decreasing function
the y-value decreases as the x-value increases
Increasing function
the y-value increases as the x-value increases
How do you tell if a Graph Represents a Function?
vertical line test
f(g(x)) =
x
Find the zeros of f(x)=x^2 −x−6.
x = 3 and x = −2
What is the domain?
x values
What is the equation to a line which passes through (-3,5) and (2, 7)?
y = 12/5x + 11/5
What is the equation to a line which has a x-intercept of (-4,0) and a y-intercept of (0,3)?
y = 3/4x+3
What is the equation to a line which is perpendicular to 3x+7y = 14 and passes through (-3, 5)?
y = 7/3x+12
What is the range?
y values
What is the equation to a line which passes through (-5,4) and has a slope of 2/3 in point slope form?
y-4 = 2/3(x+5)
point-slope form
y-y1=m(x-x1)
What is the equation to a line which is parallel to y = 4/9x + 8 and passes through (9, -16)?
y= 4/9x -20
What is vertex form?
y=a(x-h)^2+k
What is standard form?
y=ax^2+bx+c
slope intercept form
y=mx+b
Do all functions have inverses?
yes