Quiz 2 Review
1.
2.
Solutions
Graph 𝑦 = −(𝑥 − 2)2 + 3 and describe the transformations.
Graph 𝑦 = 3|𝑥 + 2| − 4 and describe the transformations.
Left 2, stretched
vertically by a factor
of 3, down 4.
Right 2, reflected
over x-axis, up 3.
3.
4.
5.
(Show at least 3 key points on each graph.)
(Show at least 3 key points on each graph.)
−√𝑥
𝑥>4
Graph 𝑓(𝑥) = {
(𝑥 + 1)2 𝑥 < 1
2 − 2𝑥 − 3 ≤ 𝑥 < 2
Graph 𝑓(𝑥) = {
3𝑥 − 2
𝑥≥2
Write the equation of each graph described:
a. 𝑦 = 𝑥 3 compressed vertically by ½ , translated left 2 and up 7
1
Answer: 𝑦 = (𝑥 + 2)3 + 7
2
b.
𝑦 = √𝑥 stretched horizontally by 3 and reflected over the x-axis and translated down 5
1
Answer: 𝑦 = −√ 𝑥 − 5
3
6.
State the domain of each function using interval notation:
a.
7.
𝑓(𝑥) =
𝑥−2
𝑥+5
(−∞, 5) ∪ (5, ∞)
b. 𝑓(𝑥) = √4𝑥 + 8
[−2, ∞)
c. 𝑓(𝑥) =
4
√3𝑥−12
(4, ∞)
State the domain and range of each function using interval notation:
a.
𝑓(𝑥) = √𝑥 + 4
D: [0, ∞) R: [4, ∞)
b. 𝑓(𝑥) = √𝑥 + 5
c. 𝑓(𝑥) = −(𝑥 − 2)2 + 3
d. 𝑓(𝑥) = 3|𝑥 + 2| − 4
D: [−5, ∞) R: [0, ∞)
D: (−∞, ∞) R: (−∞, 3]
D: (−∞, ∞) R: [−4, ∞)
𝑓(𝑥) = −4𝑥 2 + 16𝑥 − 15
a. Write f(x) in vertex form
f(x) = -4(x – 2)2 + 1
b. Find the vertex
V(2, 1)
c. Find the y intercept
(0, -15)
d. Find the x-intercepts (if it has any)
(1.5, 0) and (2.5, 0)
e. Find the max/min value
Max. of 1
f. Axis of symmetry
x=2
(−∞, 1]
g. State the range using interval notation
2
9. 𝑓(𝑥) = 𝑥 + 6𝑥 + 11
a. Write f(x) in vertex form
f(x) = (x + 3)2 + 2
b. Find the vertex
V(-3, 2)
c. Find the y intercept
(0, 11)
d. Find the x-intercepts (if it has any)
None
e. Find the max/min value
Min. of 2
f. Axis of symmetry
x = -3
[2, ∞)
g. State the range using interval notation
2
10. Solve 𝑥 + 5𝑥 − 6 ≥ 0 (write your answer using interval notation)
8.
Answer: (−∞, −6] ∪ [1, ∞) (Find x-intercepts first by solving f(x)=0. Then knowing that the graph opens upwards, we
know that the graph will be on or above the x-axis at the x-intercepts and beyond.)
11. The price-demand function for a small company that makes high-end tablets is: p(x)=1000-5x, 0  x  100, where x is the
number of items and p(x) is the price in dollars. The cost of producing and selling x items is determined by the cost
function C(x) = 500x + 4000
a. What is the price determined by a demand of 75 tablets? For 100 tablets?
p(75)=$625 per tablet
b. Determine the revenue function R(x).
R(x) = 1000x - 5x2
c. Find the revenue generated if 75 tablets are sold.
R(75) = $46,875
[0, 100]
d. What is the domain of R(x)?
e. Find the cost of making 75 tablets.
C(75) = $41,500
f. Determine the profit function P(x).
P(x) = -5x2 + 500x – 4000
g. If 75 tablets are made and sold, is the company making a profit/loss? By how much?
Profit of $5,375
h. How many items must be made and sold for the company to break even?
Solve R(x) = C(x)
1000x – 5x2 = 500x + 4000
5x2 – 500x + 4000 = 0
5(x2 – 100x + 800) = 0
Using the quadratic formula, you get: x = 8.8 or 91.3
So the company breaks even when they make and sell either 8.8 or 91.3 tablets.
12. A shot-put throw can be modeled using the equation: 𝑦 = −0.024𝑥 2 + 𝑥 + 5.5, where x is horizontal distance traveled (in
feet) and y is the height (also in feet).
a. Graph the function on your calculator and use the appropriate option to determine the maximum height that the
shot-put reaches.
This was obtained by “zooming out” several times!
Answer: The shot-put reaches a Maximum height of 15.917 feet.
b.
What is the horizontal distance traveled when it reaches its maximum height?
The x and y obtained in part a by asking the calculator for “maximum” are the coordinates of the vertex! So, the
horizontal distance when max height is obtained is the x-coordinate of the vertex.
Answer: The Horizontal distance traveled when the shot-put reaches that max height is 20.83 ft.
c.
What is the horizontal distance traveled when the shot-put to hits the ground?
This can be found by asking the calculator for the “zero” or x-intercepts (hint: you do not need the “zero” for
negative x values since that makes no sense in the problem).
Answer: The Horizontal distance when the shot-put hits the ground is 46.59 ft.
13. The table below lists the number of Americans (in thousands) who are expected to be over 100 years old for selected
years. Let x be the number of years since 1994. Use your calculator to make a scatter plot and perform a quadratic
regression on the data.
a. Write the regression equation (round coefficients to 1 decimal place)
Answer: 𝑦 = 0.4𝑥 2 + 2.0𝑥 + 50.1
b.
Use the appropriate option to predict the number of people who will be over 100 years old this year. Round your
answer to the nearest thousand.
Answer: Using the regression equation it is predicted that 251,571 Americans will be older than 100 years old in
the year 2014 (20 years since 1994).
Year
Number
(thousands)
1994
50
1996
56
1998
65
2000
75
2002
94
2004
110