Question

Compare the estimated average rate of change of the exponential function y=9(13)x and the quadratic function y=9x2+13x . Which function has a negative estimated average rate of change over the interval [0.1,0.6] ?(1 point) Responses the exponential function the exponential function neither function neither function the quadratic function the quadratic function both functions

Answer 1

To find the estimated average rate of change of the functions \( y = 9(13)x \) (exponential) and \( y = 9x^2 + 13x \) (quadratic) over the interval \([0.1, 0.6]\), we can use the formula for the average rate of change:

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

where \( a = 0.1 \) and \( b = 0.6 \).

### Step 1: Calculate for the Exponential Function

For \( y = 9(13)x \):

1. Calculate \( f(0.1) \):
\[
f(0.1) = 9(13)^{0.1}
\]

2. Calculate \( f(0.6) \):
\[
f(0.6) = 9(13)^{0.6}
\]

Now use the values:

\[
\text{Average Rate of Change} = \frac{9(13)^{0.6} - 9(13)^{0.1}}{0.6 - 0.1}
\]

### Step 2: Calculate for the Quadratic Function

For \( y = 9x^2 + 13x \):

1. Calculate \( f(0.1) \):
\[
f(0.1) = 9(0.1)^2 + 13(0.1) = 9(0.01) + 1.3 = 0.09 + 1.3 = 1.39
\]

2. Calculate \( f(0.6) \):
\[
f(0.6) = 9(0.6)^2 + 13(0.6) = 9(0.36) + 7.8 = 3.24 + 7.8 = 11.04
\]

Now use the values:

\[
\text{Average Rate of Change} = \frac{11.04 - 1.39}{0.6 - 0.1} = \frac{9.65}{0.5} = 19.3
\]

### Conclusion

Now we can summarize the results:

- **Exponential function:** The average rate of change needs the values calculated.
- **Quadratic function:** The average rate of change is \( 19.3 \) (which is positive).

Since the exponential function typically grows faster than \( x^2 \) over this interval, let’s estimate its average rate of change.

- The exponential function's average rate of change over the interval [0.1, 0.6] is positive.

Thus, the quadratic function has a positive average rate of change, and the exponential function also has a (likely positive) average rate of change.

**Final Answer:** Neither function has a negative estimated average rate of change over the interval \([0.1, 0.6]\).