Consider the function
g
(
x
)
=
13
x
2
+
6
x
−
5

A) What is the average rate of change of
g
from
14
to
20
?
B) What is the average rate of change of
g
from
16
to
20
?

We are given the function \( g(x) = 13x^2 + 6x - 5 \). To find the average rate of change of this function over an interval, we use the formula:

\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
where \( a \) and \( b \) are the endpoints of the interval.

### A) Average rate of change from 14 to 20:
Here, \( a = 14 \) and \( b = 20 \).

1. First, calculate \( g(14) \):
\[
g(14) = 13(14)^2 + 6(14) - 5 = 13(196) + 84 - 5 = 2548 + 84 - 5 = 2627
\]

2. Next, calculate \( g(20) \):
\[
g(20) = 13(20)^2 + 6(20) - 5 = 13(400) + 120 - 5 = 5200 + 120 - 5 = 5315
\]

3. Now, apply the formula for the average rate of change:
\[
\frac{g(20) - g(14)}{20 - 14} = \frac{5315 - 2627}{6} = \frac{2688}{6} = 448
\]

Thus, the average rate of change from 14 to 20 is **448**.

### B) Average rate of change from 16 to 20:
Here, \( a = 16 \) and \( b = 20 \).

1. First, calculate \( g(16) \):
\[
g(16) = 13(16)^2 + 6(16) - 5 = 13(256) + 96 - 5 = 3328 + 96 - 5 = 3419
\]

2. Use the previously calculated \( g(20) = 5315 \).

3. Now, apply the formula for the average rate of change:
\[
\frac{g(20) - g(16)}{20 - 16} = \frac{5315 - 3419}{4} = \frac{1896}{4} = 474
\]

Thus, the average rate of change from 16 to 20 is **474**.

### Do you want more details or have any questions?

Here are 5 related questions to expand on this:
1. How do you interpret the significance of average rate of change in real-world applications?
2. What would the instantaneous rate of change of \( g(x) \) be at \( x = 14 \)?
3. How does the average rate of change differ from the derivative of a function?
4. How would the average rate of change change if the interval was from 10 to 14 instead?
5. How do you calculate the rate of change for non-polynomial functions?

**Tip:** The average rate of change is closely related to the slope of the secant line between two points on a curve.

Consider the function g(x) = 13x^2 + 6x − 5. A) What is the average rate of change of g from 14 to 20? B) What is the average rate of change of g from 16 to 20?

Explore how to calculate the average rate of change for the quadratic function g(x) = 13x^2 + 6x − 5 over the intervals from 14 to 20 and from 16 to 20. Step-by-step solution provided for high school students, including key formulas and concepts in algebra.

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