Chapter 3.3, Problem 72AYU

Suppose that $f\left(x\right)=x^2+2x-8$.

(a) What is the vertex of $f$?

(b) What are the $x\text{-intercepts}$ of the graph of $f$?

(c) Solve $f\left(x\right)=-8$ for $x$. What points are on the graph of $f$?

(d) Use the information obtained in parts $(a)-(c)$ to graph $f\left(x\right)=x^2+2x-8$.

To determine

To calculate:

a. The vertex of the given function.

b. $x\text{-intercepts}$

c. Solve $f(x)=-8$ for $x$.

d. Graph the function.

Answer to Problem 72AYU

a. The vertex of the given function is $(-1,-9)$.

b. The $x\text{-intercepts}$ of the given function are at $(2,0)$ and $(-4,0)$.

c. The points on the graph of the given function are at $(0,-8)$ and $(-2,-8)$.

d. The graph of $f(x)=x^2+2x-8$ is drawn below.

Explanation of Solution

Given:

The given function is $f(x)=x^2+2x-8$.

Formula used:

For a quadratic equation of the form $f(x)=a{x}^2+bx+c,a\ne 0$, the vertex is $\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)$

Calculation:

a. Here, we have $a=1,b=2,c=-8$

$\frac{-b}{2a}=\frac{-2}{2}=-1$

$f\left(\frac{-b}{2a}\right)=f(-1)={(-1)}^2+2(-1)-8=-9$

Therefore, the vertex of the given equation will is $\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)=(-1,-9)$

b. In order to find the $x\text{-intercept}$, we have to equate the equation at $f(x)=0$ Therefore, we get

$f(x)=x^2+2x-8=0$

$\Rightarrow (x+4)(x-2)=0$

therefore,

$\Rightarrow x+4=0\Rightarrow x=-4$

and

$x - 2 = 0 \Rightarrow x = 2$

At $x=-4$, the given function is $f(-4)={(-4)}^2+2(-4)-8=0$.

At $x=2$, the given function is $f(2)={(2)}^2+2(2)-8=0$.

Thus, the $x\text{-intercepts}$ of the given function are at $(2,0)$ and $(-4,0)$.

c. Now, we have to solve $f(x)=-8$ for $x$.

$f(x)=x^2+2x-8=-8$

$\Rightarrow x^2+2x=0$

$\Rightarrow x(x+2)=0$

$\Rightarrow x=0$

$and$

$\Rightarrow x+2=0\Rightarrow x=-2$

At $x=0$, the given function is $f(-0)={(0)}^2+2(0)-8=-8$.

At $x=-2$, the given function is $f(-2)={(-2)}^2+2(-2)-8=-8$.

Thus, the points on the graph of the given function are at $(0,-8)$ and $(-2,-8)$.

d. The graph of the given function is