Question

given $f(x)=x^{3}-5$, find the equation of the secant line passing through $(-3,f(-3))$ and $(2,f(2))$. write your answer in the form $y = mx + b$.

Explanation:

Step1: Calculate $f(-3)$ and $f(2)$

First, find $f(-3)$:
\[

$$\begin{align*} f(-3)&=(-3)^3 - 5\\ &=-27 - 5\\ &=-32 \end{align*}$$

\]
Then, find $f(2)$:
\[

$$\begin{align*} f(2)&=2^3 - 5\\ &=8 - 5\\ &=3 \end{align*}$$

\]
So the two points are $(-3,-32)$ and $(2,3)$.

Step2: Calculate the slope $m$

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
Here, $x_1=-3,y_1 = - 32,x_2=2,y_2 = 3$.
\[

$$\begin{align*} m&=\frac{3-(-32)}{2-(-3)}\\ &=\frac{3 + 32}{2+3}\\ &=\frac{35}{5}\\ &=7 \end{align*}$$

\]

Step3: Find the y - intercept $b$

Use the point - slope form $y - y_1=m(x - x_1)$ with the point $(2,3)$ and $m = 7$.
\[

$$\begin{align*} y-3&=7(x - 2)\\ y-3&=7x-14\\ y&=7x-14 + 3\\ y&=7x-11 \end{align*}$$

\]

Answer:

$y = 7x-11$