Vaughan Hanson
08/13/2023 · Senior High School
Secant lines \( \& \) average rate of change What is the slope of the secant line that intersects the graph of \( g(x)=3^{2 x} \) at \( x=0 \) and \( x=2 \) ?
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Step-by-step Solution
To find the slope of the secant line that intersects the graph of \( g(x) = 3^{2x} \) at \( x = 0 \) and \( x = 2 \), we need to calculate the average rate of change of the function between these two points.
The formula for the slope of the secant line between two points \( (a, g(a)) \) and \( (b, g(b)) \) is given by:
\[
\text{slope} = \frac{g(b) - g(a)}{b - a}
\]
In this case, we have \( a = 0 \) and \( b = 2 \).
First, we calculate \( g(0) \) and \( g(2) \):
\[
g(0) = 3^{2 \cdot 0} = 3^0 = 1
\]
\[
g(2) = 3^{2 \cdot 2} = 3^4 = 81
\]
Now we can substitute these values into the slope formula:
\[
\text{slope} = \frac{g(2) - g(0)}{2 - 0} = \frac{81 - 1}{2 - 0} = \frac{80}{2} = 40
\]
Thus, the slope of the secant line that intersects the graph of \( g(x) = 3^{2x} \) at \( x = 0 \) and \( x = 2 \) is
\[
\boxed{40}
\]
Quick Answer
The slope of the secant line is 40.
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