To find the slope of the secant line between [tex]\(x = -1\)[/tex] and [tex]\(x = 1\)[/tex] on the graph of the function [tex]\(f(x) = -x^3 - 3x^2 - 2x + 1\)[/tex], follow these steps:
1. Evaluate the function at [tex]\(x = -1\)[/tex]:
Let's calculate [tex]\(f(-1)\)[/tex].
[tex]\[
f(-1) = -(-1)^3 - 3(-1)^2 - 2(-1) + 1
\][/tex]
Simplifying step by step:
[tex]\[
f(-1) = -(-1) - 3 \cdot 1 + 2 + 1
\][/tex]
[tex]\[
f(-1) = 1 - 3 + 2 + 1
\][/tex]
[tex]\[
f(-1) = 1
\][/tex]
2. Evaluate the function at [tex]\(x = 1\)[/tex]:
Let's calculate [tex]\(f(1)\)[/tex].
[tex]\[
f(1) = -(1)^3 - 3(1)^2 - 2(1) + 1
\][/tex]
Simplifying step by step:
[tex]\[
f(1) = -1 - 3 - 2 + 1
\][/tex]
[tex]\[
f(1) = -5
\][/tex]
3. Calculate the slope of the secant line:
The slope of the secant line between [tex]\(x = -1\)[/tex] and [tex]\(x = 1\)[/tex] is given by the difference in the function values at these points divided by the difference in the [tex]\(x\)[/tex]-values. The formula for the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Substituting [tex]\(x_1 = -1\)[/tex], [tex]\(x_2 = 1\)[/tex], [tex]\(f(x_1) = 1\)[/tex], and [tex]\(f(x_2) = -5\)[/tex]:
[tex]\[
m = \frac{f(1) - f(-1)}{1 - (-1)}
\][/tex]
[tex]\[
m = \frac{-5 - 1}{1 + 1}
\][/tex]
[tex]\[
m = \frac{-6}{2}
\][/tex]
[tex]\[
m = -3
\][/tex]
Therefore, the slope of the secant line between [tex]\(x = -1\)[/tex] and [tex]\(x = 1\)[/tex] on the graph of the function [tex]\(f(x) = -x^3 - 3x^2 - 2x + 1\)[/tex] is [tex]\(-3\)[/tex].
