Question

the graph of the function f(x) is given. choose the correct equation for the function from the following options:
a. $y = -(x + 1)^2(x - 2)$
b. $y = (x + 1)^2(x - 2)$
c. $y = -(x + 1)(x - 2)$
d. $y = 2(x + 1)^2(x - 2)$

Explanation:

Step1: Identify x-intercepts

The graph touches $x=-1$ (a repeated root) and crosses $x=2$, so the factored form has $(x+1)^2(x-2)$.

Step2: Determine leading coefficient

As $x\to+\infty$, $f(x)\to-\infty$, so the leading coefficient is negative.

Step3: Verify y-intercept

Substitute $x=0$ into $y=-(x+1)^2(x-2)$:

$$\begin{align*} y&=-(0+1)^2(0-2)\\ &=-(1)(-2)\\ &=2 \end{align*}$$

This matches the graph's y-intercept of 2.

Answer:

A. $y = -(x + 1)^2(x - 2)$