Question
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 1 . The average rate of change is 1. 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 13 . The average rate of change is Start Fraction 1 over 2 End Fraction . The average rate of change is 3. it starts at 0 goes up to 1 then goes up a lot more to 2
Answers
Answer
so whats the answer
Answer
hello
Answered by
GPT-4o mini
To find the average rate of change of a function \( f(x) \) over the interval \([1, 4]\), we use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 1 \) and \( b = 4 \).
From your description, it seems like the function starts at 0 at \( x=1 \) and then increases significantly by the time \( x=4 \). However, without specific values for \( f(1) \) and \( f(4) \), we can infer that if the function starts at around 0 and increases, we need to determine \( f(4) \) for accurate calculations.
Assuming from your response that \( f(1) = 0 \) and \( f(4) \) might be 2 based on your observation ("then goes up a lot more to 2"), we can plug these values into the formula:
\[
f(1) = 0, \quad f(4) = 2
\]
\[
\text{Average Rate of Change} = \frac{2 - 0}{4 - 1} = \frac{2}{3} \approx 0.67
\]
However, if we consider your other descriptions and possibilities (like the function going up to 1 or 13), you might need to find the actual values from the graph present in the image referenced.
If the given options include the average rate of change as \( \frac{1}{2} \), which implies an average increase by 0.5, we could adjust \( f(4) \) estimates accordingly.
If \( f(1) = 0 \) and \( f(4) = \frac{3}{2} \):
\[
\text{Average Rate of Change} = \frac{\frac{3}{2}-0}{4-1} = \frac{\frac{3}{2}}{3} = \frac{1}{2}
\]
Based on this general analysis, I could conclude:
- The average rate of change likely indicates the function rises overall, whether options include 1, 2, or fractional values.
- If we consider the average rate as \( \frac{1}{2} \) or perhaps errantly deduce higher or lower. If the function in your case overall increased significantly — we could statistically classify the average change as positive.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 1 \) and \( b = 4 \).
From your description, it seems like the function starts at 0 at \( x=1 \) and then increases significantly by the time \( x=4 \). However, without specific values for \( f(1) \) and \( f(4) \), we can infer that if the function starts at around 0 and increases, we need to determine \( f(4) \) for accurate calculations.
Assuming from your response that \( f(1) = 0 \) and \( f(4) \) might be 2 based on your observation ("then goes up a lot more to 2"), we can plug these values into the formula:
\[
f(1) = 0, \quad f(4) = 2
\]
\[
\text{Average Rate of Change} = \frac{2 - 0}{4 - 1} = \frac{2}{3} \approx 0.67
\]
However, if we consider your other descriptions and possibilities (like the function going up to 1 or 13), you might need to find the actual values from the graph present in the image referenced.
If the given options include the average rate of change as \( \frac{1}{2} \), which implies an average increase by 0.5, we could adjust \( f(4) \) estimates accordingly.
If \( f(1) = 0 \) and \( f(4) = \frac{3}{2} \):
\[
\text{Average Rate of Change} = \frac{\frac{3}{2}-0}{4-1} = \frac{\frac{3}{2}}{3} = \frac{1}{2}
\]
Based on this general analysis, I could conclude:
- The average rate of change likely indicates the function rises overall, whether options include 1, 2, or fractional values.
- If we consider the average rate as \( \frac{1}{2} \) or perhaps errantly deduce higher or lower. If the function in your case overall increased significantly — we could statistically classify the average change as positive.
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