A quadratic function is graphed. Compare the estimated average rate of change for the quadratic function to the estimated average rate of change for the exponential function y=−6^x

Question

on the interval [0,5]
. Use one of the symbols <
, =
, or >
to describe the comparison. the points on the graph are (0,0), (3,90), (5,90), and (8,0).

(1 point)
The estimated average rate of change of the quadratic function
the estimated average rate of change of the exponential function.

Answer 1

To compare the estimated average rates of change for the quadratic function and the exponential function $y=-6^x$ on the interval $[0,5]$, we first need to calculate the average rate of change for both functions over that interval.

1. Quadratic Function: The points on the graph you provided for the quadratic function are (0, 0), (3, 90), and (5, 90). To find the average rate of change on the interval $[0,5]$, we can use the following formula:

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

   Here, $a=0$ and $b=5$:

   $$f(0)=0,f(5)=90$$

   Plugging these values into the formula gives us:

   $$\text{Average Rate of Change for Quadratic}=\frac{90-0}{5-0}=\frac{90}{5}=18$$

2. Exponential Function: Now we consider the exponential function $y=-6^x$. However, note that the function $-6^x$ is not conventional for typical growth, as it is negative for all positive $x$. We need to evaluate it at the endpoints of the interval:

   $$y(0)=-6^0=-1$$ $$y(5)=-6^5=-7776$$

   Now, we calculate the average rate of change from $x=0$ to $x=5$:

   $$\text{Average Rate of Change for Exponential}=\frac{-7776-(-1)}{5-0}=\frac{-7776+1}{5}=\frac{-7775}{5}=-1555$$

3. Comparison: The average rate of change for the quadratic function is 18 while the average rate of change for the exponential function is -1555.

   Thus, comparing the two rates of change:

   $$18>-1555$$

The estimated average rate of change of the quadratic function is greater than the estimated average rate of change of the exponential function.

The answer is:

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