Q. 313 - Intermediate Algebra

Question

For the function, $f\left(x\right)=x^2-8x+8$,

a. Find $x$, when $f\left(x\right)=-4$.

b. Use this information to find two points that lie on the graph of the function.

Step-by-Step Solution

Answer

a. The value of $x$ is $2$or $6$when $f\left(x\right)=-4$.

b. $(2, -4)$ and $(6, -4)$ lie on the graph of the function.

Part a. Find x when f ( x ) = - 4

Substitute $f\left(x\right)=-4$ in given equation,

$$\begin{gathered} f\left(x\right)=x^2-8x+8 \\ -4=x^2-8x+8 \\ {x}^2-8x+12=0 \\ {x}^2-2x-6x+12=0 \\ x(x-2)-6(x-2)=0 \\ (x-2)(x-6)=0 \end{gathered}$$

Using Zero Product Property,

$x-2=0$ or

$x-6=0$

When $x-2=0$,

$x=2$.

When $x-6=0$,

$x=6$.

Checking each value separately,

when $x=2$,

$$\begin{gathered} f\left(x\right)=x^2-8x+8 \\ f\left(2\right)=2^2-8\left(2\right)+8 \\ f\left(2\right)=4-16+8 \\ f\left(2\right)=-4 \end{gathered}$$

This is true.

When $x=6$,

$$\begin{gathered} f\left(x\right)=x^2-8x+8 \\ f\left(6\right)=6^2-8\left(6\right)+8 \\ f\left(6\right)=36-48+8 \\ f\left(6\right)=-4 \end{gathered}$$

This is true too.

Part b. Find points that lie on the graph of the function

Since $f\left(2\right)=-4$ and $f\left(6\right)=-4$, the points $(2, -4)$and $(6, -4)$ lie on the graph of the function.