Compare the estimated average rate of change of the graphed quadratic function over the interval [2,6]

Question

to the average rate of change of the exponential function y=34(2)x
over the same interval.

(1 point)
Responses

The estimated average rate of change of the exponential function is 0.16 greater than the estimated average rate of change of the quadratic function over the interval [2,6]
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The estimated average rate of change of the exponential function is 0.16 greater than the estimated average rate of change of the quadratic function over the interval left bracket 2 comma 6 right bracket.

The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function over the interval [2,6]
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The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function over the interval left bracket 2 comma 6 right bracket.

The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the quadratic function over the interval [2,6]
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The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the quadratic function over the interval left bracket 2 comma 6 right bracket.

The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function over the interval [2,6]
.

Answer 1

To determine the average rate of change for both the quadratic function and the exponential function over the interval \([2, 6]\), let's use the average rate of change formula:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

for a function \(f(x)\) over an interval \([a, b]\).

1. For the exponential function \(y = 34(2^x)\):

Evaluate at \(x = 2\) and \(x = 6\):

• \(f(2) = 34(2^2) = 34(4) = 136\)
• \(f(6) = 34(2^6) = 34(64) = 2176\)

Now, apply the formula: \[ \text{Average Rate of Change}_{\text{exponential}} = \frac{f(6) - f(2)}{6 - 2} = \frac{2176 - 136}{4} = \frac{2040}{4} = 510 \]

2. For the quadratic function:

Since the specific quadratic function is not provided in your question, let’s assume a general form \(f(x) = ax^2 + bx + c\). For a precise answer, the specific equation would be necessary. However, if you have that function, you can follow the same process as above, finding \(f(2)\) and \(f(6)\) and applying the average rate of change formula.

Let’s assume, for example, \(f(x) = x^2\) (a simple quadratic), then:

• \(f(2) = 2^2 = 4\)
• \(f(6) = 6^2 = 36\)

Now calculate: \[ \text{Average Rate of Change}_{\text{quadratic}} = \frac{f(6) - f(2)}{6 - 2} = \frac{36 - 4}{4} = \frac{32}{4} = 8 \]

3. Compare the average rates of change:

From our calculations:

• Average rate of change of the exponential function: 510
• Average rate of change of the quadratic function: 8

Now, we can make our final conclusion: The estimated average rate of change of the exponential function is significantly greater than that of the quadratic function, particularly:

\[ 510 - 8 = 502 \]

Since the answers provided in your options do not suggest a difference as large as 502, you may need to recheck the data or choices given. If the quadratic function you are asking about is different, please provide it so I can adjust accordingly!

However, from the details provided in your question:

• The statement that best fits, assuming \(f(x) = x^2\) for the quadratic, would be: "The estimated average rate of change of the exponential function is significantly greater than the estimated average rate of change of the quadratic function over the interval [2,6]." If your answer choices must choose from existing options without this specific calculated difference, please clarify.

Answer 2

Use the image to answer the question.

A U-shaped upward opening curve with 2 plotted points is on a coordinate plane. The x-axis ranges from negative 4 to 7 in increments of 1. The y-axis ranges from negative 3 to 15 in increments of 1.

Answer 3

It seems like you are referring to an image of a U-shaped upward-opening curve (which typically suggests it is a quadratic function) with two plotted points. Unfortunately, without the actual image or further details about the points plotted on the curve or the exact nature of the question you are asking, it is challenging to provide a specific answer.

If you could provide the coordinates of the two plotted points or any specific information about the function, such as its equation or other values, I would be happy to assist you with your question.

Some possible questions you might have based on the description could include:

• Finding the average rate of change over a certain interval defined by the points.
• Analyzing the vertex or intercepts of the quadratic function.
• Comparing the quadratic function's behavior to another function (like the exponential function you mentioned previously).

Please clarify your question, and I’ll do my best to help!