Chapter 8.5, Problem 3E

To determine

ToFind:The $x\text{-intercepts}$ and axis of symmetry of the graph of the function $y=\left(x-1\right)\left(x+3\right)$

  

Answer to Problem 3E

The $x\text{-intercepts}$ of the graph of the given function are $\left(-3,0\right)$ and $\left(1,0\right)$

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

Explanation of Solution

Given:

The graph of the function $y=\left(x-1\right)\left(x+3\right)$

  

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 $y=0$

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 $y=\left(x-1\right)\left(x+3\right)$

  

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

Since, the graph of the given function touches the $x\text{-axis}$ at points $\left(-3,0\right)$ and $\left(1,0\right)$

Therefore, the $x\text{-intercepts}$ of the graph of the given function are $\left(-3,0\right)$ and $\left(1,0\right)$

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 $y=\left(x-1\right)\left(x+3\right)$

That is, $y=x^2+2x-3$

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

That is,

  $\begin{array}{c}x=-\frac{2}{1}\\ x=-2\end{array}$

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 $\left(-3,0\right)$ and $\left(1,0\right)$

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