Question
Question 1 A) Calculate the average rate of change over the interval [2,4] given the function table for f(x)=−x2+12. x f(x) 1 11 2 8 3 3 4 −4(1 point) Responses −2 negative 2 2 2 6 6 −6 negative 6 Question 2 A) Use the image to answer the question. Calculate the average rate of change over the interval [3,4] of the given graph of the function. (1 point) Responses −16 negative 16 8 8 116 Start Fraction 1 over 16 End Fraction 16 16 Question 3 A) Use the image to answer the question. Describe the graph of f(x) by selecting the correct statement. (1 point) Responses The average rate of change is negative only on the interval [1,3]. The average rate of change is negative only on the interval left bracket 1 comma 3 right bracket . The average rate of change is negative on the interval [1,3] and also on the interval [6,7]. The average rate of change is negative on the interval left bracket 1 comma 3 right bracket and also on the interval left bracket 6 comma 7 right bracket . The average rate of change is negative only on the interval [6,7]. The average rate of change is negative only on the interval left bracket six comma seven right bracket . The average rate of change is negative on the interval [0,1] and on the interval [3,4]. The average rate of change is negative on the interval left bracket 0 comma 1 right bracket and on the interval left bracket 3 comma 4 right bracket . Question 4 A) Use the image to answer the question. Given the graph of f(x), on which interval is the average rate of change the greatest? (1 point) Responses The function has the greatest average rate of change over the interval [4,5]. The function has the greatest average rate of change over the interval left bracket 4 comma 5 right bracket . The function has the greatest average rate of change over the interval [5,6]. The function has the greatest average rate of change over the interval left bracket 5 comma 6 right bracket . The function has the greatest average rate of change over the interval [1,4]. The function has the greatest average rate of change over the interval left bracket 1 comma 4 right bracket . The function has the greatest average rate of change over the interval [0,1]. The function has the greatest average rate of change over the interval left bracket 0 comma 1 right bracket . Question 5 A) Use the image to answer the question. Use the average rate of change to describe the graph of f(x) over the interval [1,4]. (1 point) Responses The average rate of change is −13. The average rate of change is negative Start Fraction 1 over 2 End Fraction . The average rate of change is 3. The average rate of change is 3. The average rate of change is 11. The average rate of change is 1. The average rate of change is 13. The average rate of change is Start Fraction 1 over 2 End Fraction .
Answers
GPT-4o mini
Let's break down each question step by step.
### Question 1 A)
To find the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\), we use the formula:
\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]
Here, \( a = 2 \) and \( b = 4 \).
First, calculate \( f(2) \) and \( f(4) \):
\[
f(2) = -2^2 + 12 = -4 + 12 = 8
\]
\[
f(4) = -4^2 + 12 = -16 + 12 = -4
\]
Now substitute these values into the formula:
\[
\text{Average rate of change} = \frac{-4 - 8}{4 - 2} = \frac{-12}{2} = -6
\]
**Response:** The average rate of change over the interval \([2, 4]\) is \(-6\).
---
### Question 2 A)
Since the graph image is not provided, we need to use general principles to calculate the average rate of change over the interval \([3, 4]\). If it represents the function you've previously described (like a quadratic), you'd find \( f(3) \) and \( f(4) \).
Assuming we have the values to plug in from the function, use the same formula:
If \( f(3) = 3 \), and \( f(4) = -4 \):
\[
\text{Average rate of change} = \frac{-4 - 3}{4 - 3} = \frac{-7}{1} = -7
\]
However, the choices are provided. Assuming the correct calculation leads to the choice option given:
**Response:** You might choose \(-16\) if it fits based on the image or data you see. Otherwise, calculate based based on graph values.
---
### Question 3 A)
If the graph indicates that the average rate of change is negative on specific intervals, select the corresponding statement based on your observation from the graph.
**Response:** Example statement would be: "The average rate of change is negative on the interval \([1, 3]\) and also on the interval \([6, 7]\)."
---
### Question 4 A)
For the interval where the average rate of change is greatest, you would typically look where the slope appears steepest on the graph.
**Response:** Choose an interval based on your observation, like: "The function has the greatest average rate of change over the interval \([1, 4]\)."
---
### Question 5 A)
To find the average rate of change over \([1, 4]\), calculate:
Assuming you found:
- \( f(1) = 11 \)
- \( f(4) = -4 \)
Plugging into the formula gives:
\[
\text{Average rate of change} = \frac{-4 - 11}{4 - 1} = \frac{-15}{3} = -5
\]
Looking at the response choices, based on graph or function values you have, choose the closest that represents this.
**Response:** It could be "The average rate of change is 3," if that closely represents the calculated slope values.
---
For accurate responses to Questions 2-5, please consult the graph images that should accompany your questions. The exact responses hinge on the values depicted therein.
### Question 1 A)
To find the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\), we use the formula:
\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]
Here, \( a = 2 \) and \( b = 4 \).
First, calculate \( f(2) \) and \( f(4) \):
\[
f(2) = -2^2 + 12 = -4 + 12 = 8
\]
\[
f(4) = -4^2 + 12 = -16 + 12 = -4
\]
Now substitute these values into the formula:
\[
\text{Average rate of change} = \frac{-4 - 8}{4 - 2} = \frac{-12}{2} = -6
\]
**Response:** The average rate of change over the interval \([2, 4]\) is \(-6\).
---
### Question 2 A)
Since the graph image is not provided, we need to use general principles to calculate the average rate of change over the interval \([3, 4]\). If it represents the function you've previously described (like a quadratic), you'd find \( f(3) \) and \( f(4) \).
Assuming we have the values to plug in from the function, use the same formula:
If \( f(3) = 3 \), and \( f(4) = -4 \):
\[
\text{Average rate of change} = \frac{-4 - 3}{4 - 3} = \frac{-7}{1} = -7
\]
However, the choices are provided. Assuming the correct calculation leads to the choice option given:
**Response:** You might choose \(-16\) if it fits based on the image or data you see. Otherwise, calculate based based on graph values.
---
### Question 3 A)
If the graph indicates that the average rate of change is negative on specific intervals, select the corresponding statement based on your observation from the graph.
**Response:** Example statement would be: "The average rate of change is negative on the interval \([1, 3]\) and also on the interval \([6, 7]\)."
---
### Question 4 A)
For the interval where the average rate of change is greatest, you would typically look where the slope appears steepest on the graph.
**Response:** Choose an interval based on your observation, like: "The function has the greatest average rate of change over the interval \([1, 4]\)."
---
### Question 5 A)
To find the average rate of change over \([1, 4]\), calculate:
Assuming you found:
- \( f(1) = 11 \)
- \( f(4) = -4 \)
Plugging into the formula gives:
\[
\text{Average rate of change} = \frac{-4 - 11}{4 - 1} = \frac{-15}{3} = -5
\]
Looking at the response choices, based on graph or function values you have, choose the closest that represents this.
**Response:** It could be "The average rate of change is 3," if that closely represents the calculated slope values.
---
For accurate responses to Questions 2-5, please consult the graph images that should accompany your questions. The exact responses hinge on the values depicted therein.