Question
Use the graph of the exponential function.
The coordinate grid has x and y axes from negative 5 to 5. An exponential function is shown.
To the nearest tenth, what is the average rate of change over the interval [−2, 2]
? What does the average rate of change indicate?
(1 point)
Responses
The average rate of change is −0.9
. On average, the y-values of the exponential function increase by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The average rate of change is negative 0 point 9 . On average, the y -values of the exponential function increase by 0 point 9 for every increase of 1 of the x -values over the interval left bracket negative 2 comma 2 right bracket .
The average rate of change is 0.9
. On average, the y-values of the exponential function decrease by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The average rate of change is 0 point 9 . On average, the y -values of the exponential function decrease by 0 point 9 for every increase of 1 of the x -values over the interval left bracket negative 2 comma 2 right bracket .
The average rate of change is 0.9
. On average, the y-values of the exponential function increase by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The average rate of change is 0 point 9 . On average, the y -values of the exponential function increase by 0 point 9 for every increase of 1 of the x -values over the interval left bracket negative 2 comma 2 right bracket .
The average rate of change is −0.9
. On average, the y-values of the exponential function decrease by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The coordinate grid has x and y axes from negative 5 to 5. An exponential function is shown.
To the nearest tenth, what is the average rate of change over the interval [−2, 2]
? What does the average rate of change indicate?
(1 point)
Responses
The average rate of change is −0.9
. On average, the y-values of the exponential function increase by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The average rate of change is negative 0 point 9 . On average, the y -values of the exponential function increase by 0 point 9 for every increase of 1 of the x -values over the interval left bracket negative 2 comma 2 right bracket .
The average rate of change is 0.9
. On average, the y-values of the exponential function decrease by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The average rate of change is 0 point 9 . On average, the y -values of the exponential function decrease by 0 point 9 for every increase of 1 of the x -values over the interval left bracket negative 2 comma 2 right bracket .
The average rate of change is 0.9
. On average, the y-values of the exponential function increase by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
The average rate of change is 0 point 9 . On average, the y -values of the exponential function increase by 0 point 9 for every increase of 1 of the x -values over the interval left bracket negative 2 comma 2 right bracket .
The average rate of change is −0.9
. On average, the y-values of the exponential function decrease by 0.9
for every increase of 1 of the x-values over the interval [−2, 2]
.
Answers
GPT-4o mini
To calculate the average rate of change of an exponential function over the interval \([-2, 2]\), we can use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \(f(x)\) is the exponential function, \(a = -2\), and \(b = 2\).
1. Determine the values of \(f(-2)\) and \(f(2)\) from the graph.
2. Substitute these values into the formula.
Assuming you found that \(f(-2)\) and \(f(2)\) from the graph are specific numerical values (let's say \(y_1\) and \(y_2\) respectively), plug those into the formula:
\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{2 - (-2)} = \frac{y_2 - y_1}{4}
\]
If, for instance, \(f(-2) = 0.1\) and \(f(2) = 2.5\):
\[
\text{Average Rate of Change} = \frac{2.5 - 0.1}{4} = \frac{2.4}{4} = 0.6
\]
In your responses, you mentioned 0.9 and -0.9 as possible values. The average rate of change indicates how much the \(y\)-values are changing for every increase of 1 of the \(x\)-values over the chosen interval. If it's positive (like 0.6), it means the function is increasing. If it's negative (like -0.9), it would indicate a decrease.
So given your options, determine which one aligns with your calculated average rate of change and interpret it correctly:
- If the average rate of change is positive, it indicates the \(y\)-values are increasing.
- If it is negative, it indicates the \(y\)-values are decreasing.
To conclude, you will need to verify the actual values from the graph to complete this calculation.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \(f(x)\) is the exponential function, \(a = -2\), and \(b = 2\).
1. Determine the values of \(f(-2)\) and \(f(2)\) from the graph.
2. Substitute these values into the formula.
Assuming you found that \(f(-2)\) and \(f(2)\) from the graph are specific numerical values (let's say \(y_1\) and \(y_2\) respectively), plug those into the formula:
\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{2 - (-2)} = \frac{y_2 - y_1}{4}
\]
If, for instance, \(f(-2) = 0.1\) and \(f(2) = 2.5\):
\[
\text{Average Rate of Change} = \frac{2.5 - 0.1}{4} = \frac{2.4}{4} = 0.6
\]
In your responses, you mentioned 0.9 and -0.9 as possible values. The average rate of change indicates how much the \(y\)-values are changing for every increase of 1 of the \(x\)-values over the chosen interval. If it's positive (like 0.6), it means the function is increasing. If it's negative (like -0.9), it would indicate a decrease.
So given your options, determine which one aligns with your calculated average rate of change and interpret it correctly:
- If the average rate of change is positive, it indicates the \(y\)-values are increasing.
- If it is negative, it indicates the \(y\)-values are decreasing.
To conclude, you will need to verify the actual values from the graph to complete this calculation.