Consider the following polynomial functions. = (2 − 7 + ) ) and = (5 − ). Which of the following statements about the polynomial functions are true? Select all that apply. o The expression that represents the sum of the polynomials is a first-degree polynomial. þ The expression that represents the difference of the polynomials illustrates that polynomials are closed under subtraction. þ The sum of () and () can be represented by the expression ) +−2. o The difference of () and () can be represented by the expression ) + − 12. þ The product of () and () can be represented by the expression − . + 3 ) + 17 − 35.
Transcript
Consider the following polynomial functions. 𝑓 𝑎 = (2𝑎 − 7 + 𝑎)) and 𝑔 𝑎 = (5 − 𝑎). Which of the following statements about the polynomial functions are true? Select all that apply. o The expression that represents the sum of the
polynomials is a first-degree polynomial. þ The expression that represents the difference of the
polynomials illustrates that polynomials are closed under subtraction.
þ The sum of 𝑓(𝑎) and 𝑔(𝑎) can be represented by the expression 𝑎) + 𝑎 − 2.
o The difference of 𝑓(𝑎) and 𝑔(𝑎) can be represented by the expression 𝑎) + 𝑎 − 12.
þ The product of 𝑓(𝑎) and 𝑔(𝑎) can be represented by the expression −𝑎. + 3𝑎) + 17𝑎 − 35.
Which of the following statements about closure is NOT true? A Polynomials are closed under addition. B Polynomials are closed under subtraction. C Polynomials are closed under multiplication. D Polynomials are closed under division.
Robin says that polynomials are closed under division. Brad adamantly disagrees. Who is correct? Justify your answer.
Brad is correct. A counterexample that shows polynomials NOT closed under division is given below.
𝒙𝟑
𝒙𝟓=𝟏𝒙𝟐 = 𝒙'𝟐
Arithmetic with Polynomials and Rational Expressions
Identify the zeros of the polynomial function 𝑁 𝑥 = $
%𝑥 − 1 𝑥 + 3 (𝑥 + 5). Show how you calculated
the zeros in the space below. Then, use the zeros to sketch the graph of the function.
𝑥 = 1, 𝑥 = −3, 𝑥 = −5
A skydiver is 760 meters above the ground when he opens his parachute. After opening the parachute, he descends at a constant speed of 17 meters per second.
Part A: What type of function would best model this situation? Explain your answer. A linear function because there is a constant rate of change.
Part B: Write the equation that models this situation.
𝒚 = 𝟕𝟔𝟎 − 𝟏𝟕𝒙, where 𝒙 represents the number of seconds after the parachute opens and 𝒚 represents the height of the parachute above the ground.
Part C: Sketch the graph that models the function.
Heig
ht o
f the
pa
rach
ute,
in
A Skydiver’s Height with respect to Time
Number of seconds after parachute is
Part D: Write an equation that would represent the number of seconds it took the skydiver to reach a height of 400 meters and use the equation to solve for that time.
𝟒𝟎𝟎 = 𝟕𝟔𝟎 − 𝟏𝟕𝒙
𝒙 ≈ 𝟐𝟏. 𝟐 The skydiver reaches 𝟒𝟎𝟎 feet at about 𝟐𝟏seconds.
Part E: What is an appropriate domain for the situation. A 𝟎 ≤ 𝒙 ≤ 𝟒𝟒. 𝟕
B 0 ≤ 𝑥 ≤ 760 C 0 ≤ 𝑦 ≤ 44.7 D 0 ≤ 𝑦 ≤ 760
Elias tossed a ball into the air. The following table represents the ball’s height with respect to time, where 𝑦 represents the ball’s height, in feet, and 𝑥 represents number of seconds after the ball was tossed.
Part A: Sketch the graph of the function.
𝑥 𝑦 0 5 0.25 8 0.5 9 0.75 8 1 5 1.25 0
Height of the ball with respect to time
Time, in seconds, after the ball is tossed
Heig
ht o
f the
bal
l, in
fe
et, a
bove
the
grou
nd
Part B: What type of function would best model this situation? Explain your answer.
A quadratic function because the second differences are constant.
Part C: Write the equation that models the situation.
𝒚 = 𝒂 𝒙 − 𝟎. 𝟓 𝟐 + 𝟗 𝟓 = 𝒂 𝟎 − 𝟎. 𝟓 𝟐 + 𝟗
−𝟒 = 𝟎. 𝟐𝟓𝒂 𝒂 = −𝟏𝟔
Vertex Form: 𝒚 = −𝟏𝟔 𝒙 − 𝟎. 𝟓 𝟐 + 𝟗 𝒚 = −𝟏𝟔(𝒙𝟐 − 𝒙 + 𝟎. 𝟐𝟓) + 𝟗 𝒚 = −𝟏𝟔𝒙𝟐 + 𝟏𝟔𝒙 − 𝟒 + 𝟗 Standard Form: 𝒚 = −𝟏𝟔𝒙𝟐 + 𝟏𝟔𝒙 + 𝟓 Part D: What is an appropriate range for the situation.
A 0 ≤ 𝑥 ≤ 1.25 B 0 ≤ 𝑥 ≤ 9 C 0 ≤ 𝑦 ≤ 1.25 D 𝟎 ≤ 𝒚 ≤ 𝟗
Part E: Kacie wrote the following one-variable equation for the situation.
𝑦 = −16 0.8 < + 16(0.8) + 5
What does Kacie’s equation represent?
The height of the ball 𝟎. 𝟖seconds after it is tossed.
In ideal conditions, 200 colony-forming units (cfu) of E. coli can grow to 400 cfu in 20minutes, to 800 cfu in 40 minutes, and to 1600 cfu in an hour.
Part A: Jenny says that the E. coli growth can be modeled with an exponential function. Do you agree? Justify your answer.
Yes, the E. coli is growing by a constant ratio.
Part B: Write the equation that models the function.
𝒚 = 𝟐𝟎𝟎 𝟖 𝒙, where 𝒚 is the cfu of E. coli and 𝒙 is the number of hours the 𝟐𝟎𝟎 cfu have been growing.
Part C: Sketch the graph of the situation.
Growth of 𝟐𝟎𝟎 cfu of E. coli
Number of hours 200 cfu of E.coli has been growing
cfu
of E
. col
i
The formula for the blood flow rate can be described by
the formula 𝐹 =𝑝1−𝑝2𝑅 , where 𝐹 represents the blood flow
rate, 𝑝) represents the pressure in the inlet, 𝑝* represents the pressure in the outlet, and 𝑅 represents the vascular resistance. Part A: Solve the formula for the vascular resistance.
Part B: Dyana says that the pressure in the outlet can be found using the formula 𝑝) = 𝑅𝐹 + 𝑝*. Is Dyana correct? Justify your answer.
No, Dyana solved for pressure in the inlet. Pressure in the outlet would be 𝒑𝟐 = 𝒑𝟏 − 𝑹𝑭.
𝑹 =𝒑𝟏−𝒑𝟐𝑭
A 25,000gallon swimming pool needs to be completely drained for maintenance purposes. Your pump empties the pool at a rate of 30gallons/minute.
Part A: Let ℎ represent the number of hours the pool has been draining. Write an equation that can be used to determine the number of hours it will take to completely drain the pool.
𝟕𝟐𝟎𝒉 = 𝟐𝟓, 𝟎𝟎𝟎
Part B: Suppose you start draining the pool Monday, May 7 at 8:00 a.m. Would it be safe to schedule maintenance to begin on Tuesday, May 8 at noon?
No, it will take over 𝟑𝟒 hours for the pool to drain. By Tuesday at noon, only 𝟑𝟐 hours have passed.
A gardener plants two types of trees in a park:
Type 𝐴 is five feet tall and grows at a rate of 12 inches per year.
Type 𝐵 is three feet tall and grows at a rate of 15 inches per year.
Algebraically determine how many years it will take for these trees to be the same height.
𝟔𝟎 + 𝟏𝟐𝒙 = 𝟑𝟔 + 𝟏𝟓𝒙 or 5+𝒙 = 𝟑 + 𝟏. 𝟐𝟓𝒙
𝒙 = 𝟖𝒚𝒆𝒂𝒓𝒔
A parking garage charges an initial fee of $7.00 for up to 2 hours of parking, and an hourly rate for each additional hour. The table below outlines the prices for up to 5 hours of parking.
Parking Rates
2 hours $7.00
3 hours $11.50
4 hours $16.00
5 hours $20.50
Which of the following linear equation(s) can be used to find 𝑥,the hourly parking rate after the initial fee?
I. 7.00 + 𝑥 = 20.50 II. 11.50 + 2𝑥 = 20.50
III. 7 + 4.50𝑥 = 16.00
IV. 3𝑥 + 7.00 = 20.50
A I only B II only C III and IV D II and IV
Liam is saving up for a new cell phone. He uses the equation 𝑃 = 𝐼 + 𝐻𝐵, where 𝑃 is the cost of the phone, 𝐼 is the amount of money he started with, 𝐻 is the number of hours he babysits, and 𝐵 is his hourly rate for babysitting. Which expression is equal to 𝐻, the number of hours Liam will need to babysit to save enough money for a cell phone?
A )
*+,
B 𝑷+𝑰𝑩
C *),
D *,)
An ice cream shop was doing research on its sales. The results showed that the relationship between the average daily temperature in Fahrenheit, 𝑡, and the daily ice cream profit in dollars, 𝑐, could be modeled by the equation 𝑐 = 17.6𝑡 − 721.6. According to the model, which of the following temperatures to the nearest tenth of a degree would make the ice cream shop have a positive profit? D
A 17.6℉ B 35.2℉ C 41.0℉ D 𝟕𝟐. 𝟐℉
LeBron attempted to solve the following equation.
12 4𝑥 − 1 =
52 + 𝑥
His work is shown below.
Part A: Based on the information provided, which
statement explains why his solution is incorrect? A
A He forgot to multiply 𝒙 by 𝟐 in Step 1. B He added 𝑥 to both sides of the equation in
Step 2 instead of subtracting 𝑥. C He added 1 on both sides of the equation in
Step 3 instead of subtracting 1. D He multiplied by the reciprocal of 3 to get 𝑥 =
2 in Step 4. Part B: Complete the following statement by circling the
correct answer.
The correct answer to the equation that LeBron
tried to solve above is 𝑥 = 1 | 𝑥 = 2 | 𝑥 = 3|𝑥 = 4.
Part B: Use the quadratic formula to solve the equation.
𝒙 = 𝟏𝟐± :𝟏𝟐 𝟐:𝟒(𝟐)(𝟓)𝟐(𝟐)
𝒙 = 𝟏𝟐± 𝟏𝟒𝟒:𝟒𝟎𝟒
𝒙 = 𝟏𝟐±𝟐 𝟐𝟔𝟒
𝒙 = 𝟏𝟐;𝟐 𝟐𝟔𝟒
or 𝒙 = 𝟏𝟐:𝟐 𝟐𝟔𝟒
𝒙 ≈ 𝟎. 𝟒𝟓 or 𝒙 ≈ 𝟓. 𝟓𝟓
Consider the quadratic equation 𝑥" − 25 = 0. Part A: Describe two methods for finding the solution to the equation. Difference of two squares; Taking the square root. Part B: Find the solution to the equation. 𝒙 − 𝟓 𝒙 + 𝟓 = 𝟎 𝒙𝟐 = 𝟐𝟓 𝒙 − 𝟓 = 𝟎or 𝒙 + 𝟓 = 𝟎 𝒙 = ± 𝟐𝟓 𝒙 = 𝟓 or 𝒙 = −𝟓 𝒙 = ±𝟓
Reasoning with Equations and Inequalities
Consider the quadratic equation 𝑥" − 9𝑥 − 36 = 0.
Part A: Mickey says the value of 𝑥 can be found using the quadratic formula. Kat says it can be found by factoring. Who is correct?
They are both correct. There are many ways to solve a quadratic equation. Part B: Find the value of 𝑥 by factoring. 𝒙 − 𝟏𝟐 𝒙 + 𝟑 = 𝟎
𝒙 − 𝟏𝟐 = 𝟎 or 𝒙 + 𝟑 = 𝟎 𝒙 = 𝟏𝟐 or 𝒙 = −𝟑
Reasoning with Equations and Inequalities
Consider the following system.
𝑦 = 4𝑥 − 1 𝑦 = −3𝑥 + 6.
Find the solution to the system using substitution. Verify the
solution by graphing.
𝟒𝒙 − 𝟏 = −𝟑𝒙 + 𝟔 𝟒𝒙 + 𝟑𝒙 − 𝟏 = −𝟑𝒙 + 𝟑𝒙 + 𝟔
𝟕𝒙 − 𝟏 + 𝟏 = 𝟔 + 𝟏 𝟕𝒙 = 𝟕 𝒙 = 𝟏
𝒚 = 𝟒 𝟏 − 𝟏
𝒚 = 𝟑 (𝟏, 𝟑)
Reasoning with Equations and Inequalities
Consider the following system of equations.
2𝑥 + 3𝑦 = 12 −4𝑥 + 5𝑦 = −4
Which of the following equations could be used to create
an equivalent system (a system with the same solution)? Select all the apply.
Determine the 𝑦 −coordinate of the solution of the system.
𝟏
Reasoning with Equations and Inequalities
The graph of the equation 𝑦 = −3𝑥 + 1 is shown below.
Based on the given information, which of the following statements are correct? Select all that apply. þ Since it lies on the graph of the line, the point (𝑎, 𝑏) is
a solution to the equation 𝑦 = −3𝑥 + 1. þ Since it does not lie on the graph of the line, the point
(𝑐, 𝑑) is not a solution of the equation 𝑦 = −3𝑥 + 1. o Since 𝑎 = 𝑐, both points (𝑎, 𝑏) and (𝑐, 𝑑) are solutions
to the equation 𝑦 = −3𝑥 + 1. þ The point /
0, − 1
0 lies on the graph of the line that
represents the equation 𝑦 = −3𝑥 + 1. þ The graph of the line represents all the solutions to the
equation 𝑦 = −3𝑥 + 1.
(𝑎, 𝑏)
(𝑐, 𝑑)
The figure below shows the graphs of two functions, 𝑓(𝑥) and 𝑔(𝑥).
Determine the positive 𝑥-coordinate where 𝑓 𝑥 = 𝑔 𝑥 .
𝑓(𝑥)
𝑔(𝑥)
𝟐
Reasoning with Equations and Inequalities
Consider the system of linear inequalities below.
𝑦 >32𝑥 − 2
𝑦 ≤ −3𝑥 − 4
Part A: Sketch the graph of the system on the coordinate plane below.
Part B: Select the box or boxes under the coordinate pairs that represent solutions for the system.
Anna will use the coordinate plane below to graph the solution set for the system of linear inequalities and the system of equations shown here.
System #1: −𝑥 + 2𝑦 ≤ 42𝑥 − 𝑦 > 6
System #2: *+𝑥 + ,
-𝑦 = 3
𝑦 = 4 − 3𝑥
Select the correct option from each drop-down box that completes the statements below. The point is the solution to System #1. The point is the solution to System #2.
(0, 0)(5, 4)(8, 6)(9, 7)
(0, 4)(1, 1)(3, −5)(6, 0)
Reasoning with Equations and Inequalities
What is the solution to the inequality 3 + #$𝑥 ≥ 1 + 𝑥 ?
A 𝑥 ≤ − *+#
B 𝑥 ≥ −7 C 𝒙 ≤ 𝟕 D 𝑥 ≥ *
$
Reasoning with Equations and Inequalities
Consider the system of equations below.
𝑥 + 3𝑦 = 5 2𝑥 + 5𝑦 = 9
Which system of equations does not have the same solution?
A −2𝑥 + 3𝑦 = −1
5𝑥 + 𝑦 = 11 B 4𝑥 + 8𝑦 = 16
2𝑥 + 𝑦 = 58 C 4𝑥 − 4𝑦 = 4
6𝑥 + 3𝑦 = 15 D 𝒙 + 𝟐𝒚 = 𝟓
−𝟑𝒙 + 𝟐𝒚 = 𝟏
Reasoning with Equations and Inequalities
On the set of axes below, draw the graph of the equation 𝑦 = − =
*𝑥 + 4.
Is the point (1,1) a solution to the equation? Explain your answer based on the graph drawn. No, the graph does not pass through the point (1,1).
Reasoning with Equations and Inequalities Nicole and Adam are each saving money for a car. The total amount of money Nicole will save is represented by the function 𝑓 𝑥 = 36 + 4𝑥. The total amount of money Adam will save is given by the function 𝑔 𝑥 = 𝑥* + 15. After how many weeks, 𝑥, will they have the same amount of money saved? Explain how you arrived at your answer.
𝟑𝟔 + 𝟒𝒙 = 𝒙𝟐 + 𝟏𝟓 𝒙 = 𝟕, 𝒙 = −𝟑
Nicole and Adam will have the same amount of money after 7 weeks. When the equations are set equal to each other, 7 is the only positive solution.
Which inequality is represented by the graph below?
A 𝑦 < #$𝑥 − 3
B 𝒚 > 𝟑𝟐𝒙 − 𝟓
C 𝑦 < −3𝑥 + 5 D 𝑦 > −2𝑥 + 3
Charles went to the hardware store and purchased a hammer, a pair of gloves, a bag of nails, and a shovel. The total cost includes the cost of the items and the state sales tax. The equation 𝑇 = 1.065𝑥 can be used to calculate the total cost of his purchase, 𝑇, where 𝑥 represents the pretax subtotal of his purchase. Part A: In the equation 𝑇 = 1.065𝑥, what does the number
"1.065" represent? D
A It represents $106.50, which is the total cost for the purchases made by Charles.
B It represents the 10.65% of state sales tax that is added to the pretax cost of the purchase.
C It represents the $1.65 state sales tax that Charles had to pay for every $10 he spent in the hardware store.
D It represents 𝟏𝟎𝟔. 𝟓%, so when multiplied by the pretax cost of the purchase, the resulting product is the total cost, including state sales tax.
Part B: Complete the following statement by writing the
correct answer in the box provided. The rate of state sales tax that Charles had to pay
is percent.𝟔. 𝟓
A ball is tossed into the air. The quadratic function ℎ 𝑡 = −16𝑡' + 14𝑡 + 5 represents the situation, where 𝑡 represents the time, in seconds, after the ball is tossed and ℎ 𝑡 represents the height of the ball, in feet. Which of the following statements is NOT true? A The constant term represents the height from which
the ball was tossed. B The factors of the function can help determine when
the ball hit the ground. C The leading coefficient indicates that the graph of the
parabola that represents the ball’s height over time will open in an upward direction.
D The graph of the parabola that represents the ball’s height over time will have a relative maximum.
Three expressions are shown below. I. 3(𝑥$ + 10𝑥 − 24) II. 3(𝑥 − 6)(𝑥 − 4) III. 3𝑥 𝑥 + 10 − 72
The expression 3𝑥$ + 30𝑥 − 72is equivalent to A I and II, only. B I and III, only.C II and III, only. D I, II, and III.
Select the expression in each row that matches the equivalent expression in each column.
(𝑥 + 2)& (𝑥 − 2)& (𝑥 + 2)(𝑥 − 2) 2(𝑥 − 1)
𝑥& − 4 o o • o
𝑥& − 4𝑥 + 4 o • o o
2𝑥& − 2𝑥𝑥 o o o •
𝑥& + 4𝑥 + 4 • o o o
The expression 𝑥" − 11𝑥 + 30 represents the function ℎ(𝑥). Part A: Factor the expression to determine the zeros of
ℎ(𝑥). 𝒉 𝒙 = (𝒙 − 𝟔)(𝒙 − 𝟓) 𝒙 = 𝟔 or 𝒙 = 𝟓
Part B: Explain what the zeros represent on the graph of
ℎ(𝑥). The zeros represent the 𝒙 −intercepts on the
graph.
Which equation has the same solutions as 𝑓 𝑥 = 3𝑥% − 2𝑥 − 5? B A 𝑓 𝑥 = (3𝑥 + 5)(𝑥 − 1) B 𝒇 𝒙 = (𝟑𝒙 − 𝟓)(𝒙 + 𝟏) C 𝑓 𝑥 = 3𝑥 + 1 𝑥 − 5
D𝑓 𝑥 = (3𝑥 − 1)(𝑥 + 5)
Find the solutions of the equation 𝑓 𝑥 = (𝑥 + 3)(𝑥 − 𝑏). D A 3 and −𝑏 B 3 and 𝑏 C −3 and −𝑏 D −𝟑 and 𝒃
Last summer, Carmelo started mowing lawns in the neighborhood to earn some money to pay for his graduation party. He mowed three lawns on Day 1 and five lawns every day after that. Which function models the total number of lawns, 𝑚(𝑑), Carmelo has mowed after 𝑑 days? A A 𝒎 𝒅 = 𝟓 𝒅 − 𝟏 + 𝟑 B 𝑚 𝑑 = 3𝑑 + 5𝑑 C 𝑚 𝑑 = 3 𝑑 − 1 + 5 D 𝑚 𝑑 = 3𝑑 − 5𝑑
Grandma Gigi gave Leyla a rare purple stone for her Sweet 16 birthday. At that time, the stone was worth $245. It has been increasing in value by the same percentage each year. When Leyla turned 21, the stone was worth $560.50. If 𝑠(𝑡) represents the value in dollars of the stone 𝑡 years after Grandma Gigi gave it to Leyla, determine the value of 𝑎 and 𝑟 if 𝑠 𝑡 = 𝑎(𝑟)0. 𝑎 = and 𝑟 =
𝟐𝟒𝟓 𝟏. 𝟏𝟖
The graph below displays 𝑔(𝑥) as a transformation of the function 𝑓 𝑥 = 𝑥'.
Which of the following functions represents 𝑔(𝑥)? A A 𝒇 𝒙 − 𝟐 − 𝟐 B 𝑓 𝑥 − 2 + 2 C 𝑓 𝑥 + 2 − 2 D 𝑓 𝑥 + 2 + 2
𝑔(𝑥)
𝑥
𝑦
Consider the following functions.
𝑓 𝑥 = 𝑥$ 𝑔 𝑥 = 2𝑥$ − 3 ℎ 𝑥 = 2𝑥* − 4𝑥 𝑚 𝑥 = 2-
𝑝 𝑥 = 𝑥$ + 2𝑥 + 1 𝑣 𝑥 = 2𝑥* − 3𝑥$ − 4𝑥 + 4
Which of the following statements are correct? Select all that apply. o 𝑓 𝑥 , 𝑔 𝑥 , and 𝑝 𝑥 are odd functions. o 𝑔(𝑥) moves 𝑓(𝑥) three units down after a vertical
stretch. o 𝑝(𝑥) represents a transformation of the independent
variable from 𝑓(𝑥). o 𝑣(𝑥) is neither an even or an odd function. o ℎ(𝑥) is an even function. o 𝑚(𝑥) is a vertical compression of 𝑓(𝑥) and a
transformation of the dependent variable from 𝑔(𝑥).
Ö Ö Ö
The graph of the equation 𝑦 = 𝑎𝑥% is shown below.
The coefficient 𝑎 is multiplied by −(
). Which of the
following statements is true? B A The graph of the new equation is wider and opens
upward. B The graph of the new equation is wider and opens
downward. C The graph of the new equation is narrower and opens
upward. D The graph of the new equation is narrower and opens
downward.
The graph of ℎ 𝑥 is shown below.
Which of the following represents ℎ 𝑥 − 2 + 3? A
A B
C D
The graph of 𝑔(𝑥) is show below.
Which of the following graphs represents 𝑔 𝑥 + 1 − 2? B
A B
C D
How does the graph of 𝑓 𝑥 = 2 𝑥 − 3 ' + 4 compare to the graph of 𝑔 𝑥 = 𝑥'? A The graph of 𝑓(𝑥) is wider than the graph of 𝑔(𝑥), and
its vertex is moved to the left 3 units and up 4 units. B The graph of 𝒇(𝒙) is narrower than the graph of 𝒈(𝒙),
and its vertex is moved to the right 𝟑 units and up 𝟒 unit.
C The graph of 𝑓(𝑥) is narrower than the graph of 𝑔(𝑥), and its vertex is moved to the left 4 units and up 3 unit.
D The graph of 𝑓(𝑥) is wider than the graph of 𝑔(𝑥), and its vertex is moved to the right 3 units and up 4 units.
Which representations are functions? I. 𝑥 𝑦
2 5
−4 9
6 −3
8 3
−4 4
III.
II. 1,2 , 4,6 , −2,6 , (5,5) IV. 𝑦 = 3𝑥 + 6
D A I only B I and IV, only C II and III, only D II, III, and IV
A function is shown in the table below.
𝑥 𝑦
−3 5
−6 −3
4 7
5 12
Which ordered pair, (−3,6) or (6, −3), would result in a relation that is not a function if included in the table? (−𝟑, 𝟔)
The function 𝑔 has a domain of {−2, 0, 4, 13} and a range of {5,9,15}. Could 𝑔 be represented by the following relation: { −2, 5 , 0, 9 , 4, 15 , 13, 5 }? Justify your answer. Yes, every element in the domain corresponds to only one element in the range.
If 𝑟 𝑠 = 𝑠 + 2 & − 4𝑠, which statement is true? A 𝒓 𝟐 = 𝟖
B 𝑟 4 = 18 C 𝑟 −1 = −3 D 𝑟 −10 = 24
The first term of a sequence is 9 and the common difference is 5. Which of the following is the fifth term of the sequence? D
A −11 B −6 C 24 D 𝟐𝟗
During a tropical thunderstorm, a meteorologist recorded the accumulating rainfall over a seven-hour period. During the first four hours of the thunderstorm, the rain accumulated at a constant rate of one-half inch per hour. The rain stopped for the next hour and then started again at a constant rate of 2 inches per hour for the next two hours. Represent the situation as a graph below.
Number of Hours
Acc
umul
ated
Rai
nfal
l (in
ches
)
Consider the graph below.
Which interval has the greatest rate of change? A A 𝟎 ≤ 𝒙 ≤ 𝟓
B 5 ≤ 𝑥 ≤ 10 C 10 ≤ 𝑥 ≤ 15 D 15 ≤ 𝑥 ≤ 20
Sketch the graph of the function ℎ 𝑥 = 𝑥$ − 10𝑥 + 22 on the coordinate plane below. Circle the zeros and draw a box around the vertex.
Which function and ordered pair represent the equivalent vertex form and vertex for the function 𝑔 𝑥 = 𝑥$ − 4𝑥 + 5?
A 𝑥 − 2 $ + 1, (−2, 1) B 𝒙 − 𝟐 𝟐 + 𝟏, (𝟐, 𝟏) C 𝑥 − 2 $ + 9, (−2,9) D 𝑥 − 2 $ + 9, (2,9)
Which quadratic function has the smallest minimum? A
A 𝑓 𝑥 = (𝑥 − 3)(𝑥 + 1)
C 𝑔 𝑥 = 2𝑥- − 4𝑥 − 1
B 𝑥 𝑦
−1 5
0 2
1 1
2 2
3 5
D
Graph the following function on the coordinate plane below.
𝑓 𝑥 = 𝑥 − 4 , −1 ≤ 𝑥 < 64,6 ≤ 𝑥 < 8
Consider the following function rules.
𝑓 𝑥 = −4𝑥 + 10
𝑔 𝑥 =34𝑥
+ − 1
Determine if the following statements are true or false.
Statement True False
𝑓 2 = 𝑔(2) • o
𝑓 1 = 6 and 𝑔 1 = −01 • o
𝑓 0 = 10 and 𝑔 0 = 1 o • 𝑓 3 + 𝑔 3 = 3.75 • o
𝑓 −2 = 𝑔(−2) o • 𝑓 7 + 𝑔 5 =
14 o •
𝑓 4 = −6and 𝑔 4 = 11 • o
Consider the function 𝑓 𝑥 = 𝑥$ − 4𝑥 − 4 and the graph of the function 𝑔(𝑥) sketched in the figure below.
Based on the given information, which of the following statements are true? Select all that apply. o 𝑓(𝑥) and 𝑔(𝑥) are functions because each of them
assigns a single unique output for each of their inputs. o The function 𝑓(𝑥) has more elements in its range than
the function 𝑔(𝑥). o The values −8,−3, 0, 2, 9 are elements of the range
for both functions 𝑓(𝑥) and 𝑔(𝑥). o The axis of symmetry of 𝑓(𝑥) is the same as the axis of
symmetry of 𝑔(𝑥). o The domain of the function 𝑓(𝑥) is all real numbers,
which is the same domain of function 𝑔(𝑥).
𝑥
𝑦
𝑔(𝑥)
Ö
Ö
Ö
The restaurant Pizza by the 𝜋 offers six sizes: personal (8 in), small (10 in), medium (12 in), large (14 in), extra-large (16 in), and 𝜋-zilla (24 in). The table below shows the cost of a cheese pizza as a function of the diameter of the pizza.
Part A: What is the average rate of change in cost when
moving from a medium pizza to a 𝜋-zilla? B
A $0.79 B $𝟏. 𝟐𝟓 C $1.49 D $1.99
Part B: What does this value mean in the context of this
problem?
On average, for each 1-inch increase in diameter (from the medium to the pi-zilla), the cost of the pizza increases by $𝟏. 𝟐𝟓.
Suppose a rocket is launched from a platform. The function ℎ(𝑡) = −4.9𝑡* + 200𝑡 + 25 gives the rocket’s height,ℎ, in meters, in terms of time, 𝑡, in seconds. The function ℎ(𝑡) is equivalent to −4.9 𝑡 − 20.41 * + 2,065.82. Respond to Part A, Part B, and Part C based on the given information.
Part A: Select the answer that completes the following
statement.
The parabola opens
Part B: Complete the following statements by writing the correct answer in the boxes below.
The initial height at which the rocket is launched is
meters.
The initial velocity of the rocket is m/s.
Part C: Identify the vertex of the function ℎ(𝑡). Round to the nearest hundredth.
, 𝟐𝟎. 𝟒𝟏 𝟐𝟎𝟔𝟓. 𝟖𝟐
down. up.
𝟐𝟓
𝟐𝟎𝟎
Draw a line to connect the statement that best describes each function.
It has no real roots. 𝑓 𝑥 = 𝑥 + 4 & − 9
The vertex is located in
Quadrant III.
𝑓 𝑥 = 4𝑥& − 3
The𝑦-intercept is located at the
origin.
𝑓 𝑥 = −𝑥(𝑥 − 7)
It has two zeros, both located
between −1 and 1. 𝑓 𝑥 = −2𝑥& + 3𝑥 − 7
Given the following quadratic functions
𝑓 𝑥 = 𝑥$ − 6𝑥 − 16
and
𝑥 −2 −1 0 1 2 3 4 5
𝑔(𝑥) −7 −12 −15 −16 −15 −12 −7 0
Which statement about these functions is true?
A The vertex of 𝑔(𝑥) and the vertex of 𝑓(𝑥) are both located in quadrant III.
B The 𝑦 −intercept of 𝑓(𝑥) is greater than the 𝑦 −intercept of 𝑔(𝑥).
C The function 𝑔(𝑥) has a greater maximum value than 𝑓 𝑥 .
D The sum of the roots of 𝑓 𝑥 is greater than the sum of the roots of 𝑔 𝑥 .
Which of the following is a tabular representation of a linear function? C
A
𝑥 𝑓(𝑥) 1 15 2 17 3 21 4 27
C 𝑥 𝑔(𝑥) 1 −8 2 −6 3 −4 4 −2
B 𝑥 ℎ(𝑥) 1 3 2 6 3 12 4 24
D 𝑥 𝑘(𝑥) 1 10 2 15 3 25 4 40
Dr. León is a chiropractor who offers house calls to his patients. He uses the function 𝑐 𝑚 = 26𝑚 + 60 to calculate the total medical bill (in dollars), 𝑐 𝑚 , for each 20-min session, 𝑚. During the holidays, Dr. León changes his total medical bill function to 𝑐 𝑚 = 35𝑚 + 45. Which of the following statements are correct? Select all that apply. o Dr. León’s usual charge for a 20-min session is $35. o Dr. León charges a $45 flat fee for making the house
call during the holidays. o Dr. León charges $35 per 20-min session worked
during the holidays. o Dr. León usually charges a $26 flat fee for making the
house call. o Dr. León drops his house call flat fee by $15during the
holidays. o Dr. León drops his fee by $9for each 20-min session
during the holidays. o Dr. León charges $138 for 1-hour session any time of
the year, but during the holidays he charges $12 more.
Ö Ö
Ö
Ö
Consider the following scenarios.
#1: After reading three books this year, Dwayne joined a book club, and now he reads two books every month.
#2: The volume of a landfill doubles every three years.
Part A: Complete the following statements by selecting and circling the correct option.
Scenario #1 can be modeled by a(n)
function.
Scenario #2 can be modeled by a(n)
function.
Part B: Which of the following statements is correct? B
A The function that models scenario #1 grows by equal differences over decreasing intervals.
B The function that models scenario #2 grows by equal factors over equal intervals.
C The rate of change is the same in both scenarios.
D In both scenarios, we see quantities doubling each time.
exponential linear
quadratic
exponential linear
quadratic
Which situation(s) could by modeled by a linear function? I. You have an assignment due and your teacher
deducts 2 points for each day the assignment is turned in after the due date.
II. A car depreciates at a rate of 15% per year. III. A bicyclist travels at a constant rate of 25 miles per
hour for 30 minutes. She stops for 10 minutes and then travels at a constant rate of 30 miles per hour for 15 minutes.
A I, only B II, only C III, only D I and III, only
A culture of 100 bacteria triples every hour. Which function represents, ℎ(𝑡), the culture after 𝑡 hours? D A ℎ 𝑡 = 100𝑡 + 3 B ℎ 𝑡 = 3𝑡 + 100 C ℎ 𝑡 = 3 100 * D 𝒉 𝒕 = 𝟏𝟎𝟎 𝟑 𝒕
Consider the pattern of triangles shown below:
Should a linear or exponential function be used to determine the number of triangles in the 𝑛"# model? Justify your answer. Exponential. The number of triangles is doubling each time.
As 𝑥 increases beyond 45, which function will have the largest value? B A ℎ 𝑡 = 1.3𝑥 B 𝒉 𝒕 = 𝟏. 𝟑𝒙 C ℎ 𝑡 = 1.3𝑥/ D ℎ 𝑡 = 1.3𝑥0 + 1.3𝑥/
Write a function rule for the following sequences. Use the notation provided in the table. Term Number 1 2 3 4 … 𝑛Term 17 13 9 5 … 𝑓(𝑛)
𝒇 𝒏 = −𝟒𝒏 + 𝟐𝟏
A cell phone company charges a one-time activation fee, as well as a monthly service charge for a cell phone plan with unlimited minutes and data. The total cost is modeled by the function 𝑦 = 99 + 85𝑥. Which statement represents the meaning of the function?
A 𝑦 is the total cost, 𝑥 is the number of months of service, $85 is the one-time activation fee, and $99 is the service charge per month.
B 𝒚 is the total cost, 𝒙 is the number of months of service, $𝟗𝟗 is the one-time activation fee, and $85 is the service charge per month.
C 𝑥 is the total cost, 𝑦 is the number of months of service, $99 is the one-time activation fee, and $85 is the service charge per month.
D 𝑥 is the total cost, 𝑦 is the number of months of service, $85 is the one-time activation fee, and $99 is the service charge per month.
Examine the two figures below.
Figure A Figure B
Let 𝑎 be a non-zero, rational number and 𝑏 be an irrational number. Which of the following statements are correct? Select all that apply. o The area of Figure A is an irrational number. o The perimeter of Figure A is an irrational number. o The area of Figure B is an irrational number. o The perimeter of Figure B is a rational number. o The sum and the product of the two 𝑎-unit long sides
of Figure B is a rational number. o The sum and the product of the two 𝑏-unit long sides
of Figure B is a rational number. o The product of the areas of both figures is a rational
number.
√3
𝑎
√3 𝑏
Ö Ö Ö
Consider the following expressions.
I. 121 ∙ 49 II. &
'+ )
&*
III. &+∙ 5
IV. 2 81
Which expression(s) result in an irrational number? B A II, only B III, only C I, II, IV D II, III, IV
Evaluate the following expression.
15
#∙ 250#
𝟓𝟎𝟑
Which of the following is equivalent to 𝑥"#?
A 𝑥$#
#
B (𝑥&)( C 𝑥" (
D None of the above are equivalent
Which of the following statements is not always true?
A The product of a nonzero rational number and irrational number is irrational.
B The product of two irrational numbers is irrational. C The sum of two rational numbers is rational. D The sum of a rational number and an irrational
number is irrational.
A group of college students participated in a study in which they sampled four smart phones: iPhone 7, Galaxy S7, Moto Z, and Pixel. They tried each phone for a week then, after the four-week trial, they were each asked to select their favorite phone. The results of the survey are shown in the table below. iPhone 7 Galaxy S7 Moto Z Pixel
Males 20 30 5 5Females 25 20 5 10
Based on the survey results, which statements are valid? Select all that apply.
o About 21% of the females preferred the iPhone 7.
o ()*
of the college students preferred the Galaxy S7.
o About 8.3% of the college students preferred the
Moto Z.
o )*-
of the males preferred the Pixel.
o One-third of the males preferred the iPhone 7.
o About one-sixth of the females preferred the Moto Z.
Ö Ö
Ö
The graph below displays data on the profit Tommy makes at his candy bar stand.
Part A: Complete the following statement. Tommy earns for each candy bar that he sells.
Part B: The regression equation 𝑦 = 𝐴 + 𝐵𝑥 can be used to model the data displayed in the graph. What do the variables 𝐵, 𝑥, and 𝑦 represent in the equation?
Profit ($) Number of candy bars
Slope of the regression line
𝐵 o o • 𝑥 o • o 𝑦 • o o
Number of candy bars sold
Prof
it ($
)
Candy Bar Profit
$𝟎. 𝟓𝟎
Two business partners, Juan and Nelson, are looking to rent an office. They are interested in renting an 800-square-foot office that is listed for$1,325 a month. They conducted some research on eight commercial buildings that were rented within the past month near the location where they would like to rent. They collected information on office size (in square feet) and monthly rent (in dollars).
Part A: Using the model 𝑦 = 𝐴 + 𝐵𝑥, compute the regression
equation for the data in the table. Write the values for 𝐴 and 𝐵 into the text boxes in the equation below.
𝑦 = +𝑥
Part B: Juan concluded that there is a positive relationship
between office size and the cost of rent. Nelson agrees and added that the office space they are interested in renting is overpriced. Which of the business partners is correct? C
The dot plots below show the ages of the members of two soccer teams that play in a local community league which aims to motivate youth through sports.
Based on the information presented, complete the statements on the following page by selecting the correct answer in each box.
Age of Soccer Players in Team A
Age of Soccer Players in Team B
The median age of the players on Team A is
the median age of the players on B.
The mean age of the players on Team A is
the mean age of the players on B.
The IQR for the age of the players on Team A is
the IQR for the age of the players on B.
The first quartile of players’ age for Team A is
the first quartile of players’ age for B.
greater than equal to less than
greater than equal to less than
greater than equal to less than
greater than equal to less than
Consider the following table that compares the depth of the Columbia River to the velocity at that depth.
Part A: Use technology to determine the correlation
coefficient of the data. −𝟎. 𝟗𝟑𝟖𝟐
Part B: What does the correlation coefficient tell you in terms of the context?
There is a strong negative correlation, so the deeper the
river is, the slower the velocity is.
The student government association sold hot chocolate
and bowls of chili at their concession stand. The treasurer recorded the average number sold each week. She found the line of best fit and correlation coefficient, displayed below.
Which of the following statement(s) can you conclude from the given data. I. Drinking hot chocolate causes someone to become
hungry. II. Eating chili causes someone to become thirsty. III. There is a strong correlation between sales of hot
