Chapter 8.5, Problem 5E

To determine

ToFind:The $x\text{-intercepts}$ and axis of symmetry of the graph of the function $f\left(x\right)=-5\left(x+7\right)\left(x-5\right)$

Answer to Problem 5E

The $x\text{-intercepts}$ of the graph of the given function are $\left(-7,0\right)$ and $\left(5,0\right)$

The axis of symmetry of the graph of the given function is $x=-2$

Explanation of Solution

Given:

The graph of the function $f\left(x\right)=-5\left(x+7\right)\left(x-5\right)$

Concept Used:

The $x\text{-intercept}$ of the graph of a function is that point on the graph of the function at which the function touches the $x\text{-axis}$.

That is, it is that point on the graph of the function at which $y=f\left(x\right)=0$

The axis of symmetry of the graph of a function is that line which divides the graph into two concurrent halves.

The axis of symmetry of a parabola with equation $y=a{x}^2+bx+c$ is the vertical line given by $x=-\frac{b}{a}$

Calculation:

Given the graph of the function $f\left(x\right)=-5\left(x+7\right)\left(x-5\right)$

The $x\text{-intercept}$ of the graph of a function is that point on the graph of the function at which the function touches the $x\text{-axis}$.

That is, it is that point on the graph of the function at which $y=f\left(x\right)=0$

Substituting $f\left(x\right)=0$ in the given function $f\left(x\right)=-5\left(x+7\right)\left(x-5\right)$

  $\begin{array}{c}f\left(x\right)=-5\left(x+7\right)\left(x-5\right)\\ 0=-5\left(x+7\right)\left(x-5\right)\end{array}$

That is

  $x+7=0$

and

  $x-5=0$

  $x=-7$

and

  $x=5$

Thus, $f\left(x\right)=0$ at the points $\left(-7,0\right)$ and $\left(5,0\right)$

Therefore, the $x\text{-intercepts}$ of the graph of the given function are $\left(-7,0\right)$ and $\left(5,0\right)$

The axis of symmetry of a parabola with equation $y=a{x}^2+bx+c$ is the vertical line given by $x=-\frac{b}{a}$

For the graph of the function $f\left(x\right)=-5\left(x+7\right)\left(x-5\right)$

That is,

  $\begin{array}{c}f\left(x\right)=-5\left(x^2+2x-35\right)\\ f\left(x\right)=-5x^2-10x+175\end{array}$

The axis of symmetry is given by $x=-\frac{b}{a}$

That is,

  $\begin{array}{c}x=\frac{-\left(-10\right)}{-5}\\ x=-2\end{array}$

Therefore, the axis of symmetry of the graph of the given function is $x=-2$

Conclusion:

The $x\text{-intercepts}$ of the graph of the given function are $\left(-7,0\right)$ and $\left(5,0\right)$

The axis of symmetry of the graph of the given function is $x=-2$