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

neither function
neither function

the quadratic function
the quadratic function

both functions
both functions

the exponential function

Answer 1

To compare the estimated average rate of change of the two functions \(y = 9(13)x\) (exponential) and \(y = 9x^2 + 13x\) (quadratic) over the interval \([0.1, 0.6]\), we first need to calculate the values of each function at the endpoints of the interval.

### Step 1: Calculate the function values at the endpoints

**For the exponential function:**
\[
y = 9(13)x
\]
- At \(x = 0.1\):
\[
y(0.1) = 9(13)(0.1) = 9(1.3) = 11.7
\]

- At \(x = 0.6\):
\[
y(0.6) = 9(13)(0.6) = 9(7.8) = 70.2
\]

**For the quadratic function:**
\[
y = 9x^2 + 13x
\]
- At \(x = 0.1\):
\[
y(0.1) = 9(0.1^2) + 13(0.1) = 9(0.01) + 1.3 = 0.09 + 1.3 = 1.39
\]

- At \(x = 0.6\):
\[
y(0.6) = 9(0.6^2) + 13(0.6) = 9(0.36) + 7.8 = 3.24 + 7.8 = 11.04
\]

### Step 2: Calculate the average rate of change

The average rate of change over the interval \([a, b]\) is given by the formula:
\[
\frac{f(b) - f(a)}{b - a}
\]

**For the exponential function:**
\[
\text{Average Rate} = \frac{y(0.6) - y(0.1)}{0.6 - 0.1} = \frac{70.2 - 11.7}{0.5} = \frac{58.5}{0.5} = 117
\]

**For the quadratic function:**
\[
\text{Average Rate} = \frac{y(0.6) - y(0.1)}{0.6 - 0.1} = \frac{11.04 - 1.39}{0.5} = \frac{9.65}{0.5} = 19.3
\]

### Step 3: Analyze the rates

- The average rate of change for the exponential function over the interval \([0.1, 0.6]\) is **117**.
- The average rate of change for the quadratic function over the interval \([0.1, 0.6]\) is **19.3**.

### Conclusion

Both average rates of change are positive. Thus, neither function has a negative estimated average rate of change over the interval \([0.1, 0.6]\).

**Final Response: neither function**