Compare the slopes of the linear functions f(x) and g(x) and choose the answer that best describes them.

Question

a line labeled f of x passing through 0, negative 1 and 3, 1	
x	g(x)
0	3
2	6
4	9
 The slope of f(x) is greater than the slope of g(x).
 The slope of f(x) is less than the slope of g(x).
 The slope of f(x) is equal to the slope of g(x).
 The slope of g(x) is undefined.
Question 2(Multiple Choice Worth 4 points)
(02.02 MC)

A group of friends go to the movies. The function h(x) represents the amount of money spent, where x is the number of friends at the movies. Does a possible solution of (6.5, $95.25) make sense for this function? Explain your answer.

Yes. The input and output are both possible.
 No. The input is not possible.
 No. The output is not possible.
 No. Neither the input nor output is possible.
Question 3(Multiple Choice Worth 4 points)
(02.05 MC)

The table represents an absolute value function f(x).

x	f(x)
−5	1
−4	0
−3	1
−2	2
−1	3
0	4
1	5
2	6
3	7

What are the vertex and range of the function?
 Vertex (0, 4), Range: {y | 0 ≤ y < ∞}
 Vertex (0, 4), Range: {y | 4 ≤ y < ∞}
 Vertex (−4, 0), Range: {y | 0 ≤ y < ∞}
 Vertex (−4, 0), Range: {y | −4 ≤ y < ∞}
Question 4(Multiple Choice Worth 4 points)
(02.03 MC)

The graph shows the distance (y) between Hannah and her home, in km, after certain amounts of time (x), in hours.

Graph of first line going through ordered pairs 0, 4 and 2, 4. Graph of second line going through ordered pairs 2, 4 and 3, 2. Graph of third line going through ordered pairs 3, 2 and 4, 4. Graph of fourth line going through ordered pairs 4, 4 and 6, 4.

Which of the following statements best describes Hannah's distance from her home?

It is decreasing in the interval 2 < x < 4 hours.
 It is increasing in the interval 2 < x < 3 hours.
 It is decreasing in the interval 2 < x < 3 hours.
 It is increasing in the interval 2 < x < 4 hours.
Question 5(Multiple Choice Worth 4 points)
(02.04 MC)

Write the equation of the line that passes through the points (1, 7) and (5, 15) using function notation.

y = 2x + 5
 y = 4x + 8
 f(x) = 2x + 5
 f(x) = 4x + 8
Question 6(Multiple Choice Worth 4 points)
(02.04 MC)

What is the equation of the line that is perpendicular to y = 4x + 6 and passes through the point (8, −4)?

y equals negative one-fourth times x minus 2
 y equals negative one-fourth times x plus 7
 y = 4x − 36
 y = 4x + 24
Question 7(Multiple Choice Worth 4 points)
(02.05 LC)

Which of the following statements best describes the effect of replacing the graph of y = f(x) with the graph of y = f(x) − 9?

The graph of y = f(x) will shift up 9 units.
 The graph of y = f(x) will shift down 9 units.
 The graph of y = f(x) will shift left 9 units.
 The graph of y = f(x) will shift right 9 units.
Question 8(Multiple Choice Worth 4 points)
(02.05 MC)

What is the range of f(x) = |x| − 5?

−∞ < y ≤ −5
 −5 ≤ y < ∞
 0 ≤ y < ∞
 5 ≤ y < ∞
Question 9(Multiple Choice Worth 4 points)
(02.03 MC)

Choose the graph that correctly represents the equation 3x + 9y = −18.

line through the points negative 6 comma 0 and 0 comma 2
 line through the points negative 6 comma 0 and 0 comma negative 2
 line through the points negative 2 comma 0 and 0 comma 6
 line through the points negative 2 comma 0 and 0 comma negative 6
Question 10(Multiple Choice Worth 4 points)
(02.01, 02.03 LC)

Which set represents the domain of the function shown?

{(−3, 6), (0, 2), (4, 7), (11, 15)}

{(6, −3), (2, 0), (7, 4), (15, 11)}
 {2, 6, 7, 15}
 {−3, 0, 4, 11}
 {−3, 0, 2, 4, 6, 7, 11, 15}
Question 11(Multiple Choice Worth 4 points)
(02.02 LC)

If h(x) = −4x − 10, find h(−5).

−30
 1.25
 10
 −1
Question 12(Multiple Choice Worth 4 points)
(02.02 MC)

Given the function g(x) = −3x + 4, compare and contrast g(−2) and g(4). Choose the statement that is true concerning these two values.

The value of g(−2) is smaller than the value of g(4).
 The value of g(−2) is the same as the value of g(4).
 The values of g(−2) and g(4) cannot be compared.
 The value of g(−2) is larger than the value of g(4).
Question 13(Multiple Choice Worth 4 points)
(02.03 HC)

The following table shows the amount of air leaking from an inflatable as a function of time:

Time (in minutes)
x	Air (in ounces)
f(x)
0	64
1	56
2	48
3	40
4	32

Find and interpret the meaning of the x-intercept in this scenario.
 (8, 0); the time it takes all the air to leave the inflatable
 (64, 0); the time it takes all the air to leave the inflatable
 (64, 0); the time it takes to fill up inflatable with air
 (8, 0); the time it takes to fill up inflatable with air
Question 14(Multiple Choice Worth 4 points)
(02.05 LC)

Which of the following statements best describes the effect of replacing the graph of y = f(x) with the graph of y = f(x − 2)?

The graph of y = f(x) will shift left 2 units.
 The graph of y = f(x) will shift right 2 units.
 The graph of y = f(x) will shift up 2 units.
 The graph of y = f(x) will shift down 2 units.
Question 15(Multiple Choice Worth 4 points)
(02.04 MC)

A medical clinic is reducing the number of incoming patients by giving vaccines before flu season. During week 5 of flu season, the clinic saw 90 patients. In week 10 of flu season, the clinic saw 60 patients. Assume the reduction in the number of patients each week is linear. Write an equation in function form to show the number of patients seen each week at the clinic.

f(x) = −6x + 120
 f(x) = 6x + 120
 f(x) = 30x + 90
 f(x) = −30x + 90

GPT-4o mini · Answer

Here are the answers to the questions: **Question 1:** To find the slopes of the functions $ f(x) $ and $ g(x) $: - For $ f(x) $, the points $ (0, -1) $ and $ (3, 1) $ give the slope: $$ \text{slope} = \frac{1 - (-1)}{3 - 0} = \frac{2}{3} $$ - For $ g(x) $, the points given in the table are: - $ (0, 3) $ and $ (2, 6) $ $$ \text{slope} = \frac{6 - 3}{2 - 0} = \frac{3}{2} $$ Since $ \frac{2}{3} < \frac{3}{2} $: **Answer:** The slope of $ f(x) $ is less than the slope of $ g(x) $. --- **Question 2:** The input of 6.5 friends doesn't make sense as we cannot have a fractional number of friends attending: **Answer:** No. The input is not possible. --- **Question 3:** The vertex of the given absolute value function $ f(x) $ appears to be at $ (-4, 0) $ based on the values given, since $ f(-4) = 0 $ and the function increases from there: - Range: The minimum value is $ 0 $ and increases indefinitely. **Answer:** Vertex (−4, 0), Range: {y | 0 ≤ y < ∞}. --- **Question 4:** In the interval $ 2 < x < 4 $, Hannah's distance shows she is moving away from home at $ y= 4 $ until she decreases in distance after 3 hours implying she is returning home: **Answer:** It is decreasing in the interval 2 < x < 4 hours. --- **Question 5:** To find the slope between the points (1, 7) and (5, 15): Slope $ m = \frac{15 - 7}{5 - 1} = \frac{8}{4} = 2 $. Thus, the equation in slope-intercept form is: $$ y - 7 = 2(x - 1) \rightarrow y = 2x + 5 $$ **Answer:** $ f(x) = 2x + 5 $. --- **Question 6:** The slope of the line $ y = 4x + 6 $ is 4; therefore, the slope of the perpendicular line will be $ -\frac{1}{4} $. Using point-slope form with point (8, -4): $$ y + 4 = -\frac{1}{4}(x - 8) \rightarrow y = -\frac{1}{4}x + 2 $$ **Answer:** $ y = -\frac{1}{4}x + 2 $. --- **Question 7:** Revising $ y = f(x) - 9 $ results in the graph being shifted downwards: **Answer:** The graph of $ y = f(x) $ will shift down 9 units. --- **Question 8:** Since the absolute value function $ f(x) = |x| $ has a minimum value of 0, translating it down 5 yields: **Answer:** $ -5 ≤ y < ∞ $. --- **Question 9:** To find the intercepts: - Set $ y = 0 $: $ 3x = -18 \rightarrow x = -6 $. - Set $ x = 0 $: $ 9y = -18 \rightarrow y = -2 $. Thus, the line goes through $ (-6, 0) $ and $ (0, -2) $: **Answer:** line through the points -6, 0 and 0, -2. --- **Question 10:** The domain of the function is represented by the x-coordinates: **Answer:** {−3, 0, 4, 11}. --- **Question 11:** Substituting into $ h(x) = -4(-5) - 10 $: $$ h(-5) = 20 - 10 = 10 $$ **Answer:** 10. --- **Question 12:** Calculate $ g(-2) $ and $ g(4) $: $$ g(-2) = 10, \quad g(4) = -8 $$ Thus, $ g(-2) > g(4) $: **Answer:** The value of $ g(-2) $ is larger than the value of $ g(4) $. --- **Question 13:** The x-intercept where the amount of air is 0 indicates the time until all air is leaked: **Answer:** (8, 0); the time it takes all the air to leave the inflatable. --- **Question 14:** Revising $ y = f(x-2) $ translates to the right: **Answer:** The graph of $ y = f(x) $ will shift right 2 units. --- **Question 15:** Finding the slope with $ y = mx + b $: The slope from 90 to 60 reduces over 5 weeks: $$ m = \frac{60 - 90}{10 - 5} = -6 $$ With the y-intercept $ b = mx + b$, at $ x = 5, y = 90 $: $$ 90 = -6(5) + b \rightarrow b = 120 $$ The function becomes: **Answer:** $ f(x) = -6x + 120 $. Feel free to ask if you need clarification on any specific answer!