Chapter 1.2, Problem 59AYU

In Problems 49-64, list the intercepts and test for symmetry.

$y=x^2-3x-4$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 59AYU

$x\text{-intercept}$ are $-1$ and 4 and $y\text{-intercept}$ is $-4$. The graph of the equation is not symmetric with respect to $x\text{-axis}$, $y\text{-axis}$ and the origin.

Explanation of Solution

Given:

The equation $y=x^2-3x-4$

To find the $x\text{-intercept(s)}$, let $y=0$

Therefore                           $0=x^2-3x-4$

$x^2-3x-4=0$

$(x^2-4x)+(x-4)=0$   Splitting the middle term and grouping the terms

$x\left(x-4\right)+1\left(x-4\right)=0$   Taking common factor $x$ from first two terms

$\left(x-4\right)\left(x+1\right)=0$

$x-4=0$ or $x+1=0$

$x=4$ or $x=-1$

The $x\text{-intercept}$ are 4 and $-1$

To find the $y\text{-intercept(s)}$, let $x=0$

Therefore $y={(0)}^2-3(0)-4$

$y=-4$

The $y\text{-intercept}$is $-4$

Now test the equation for symmetry with respect to $x\text{-axis}$. Replace $y$ by $-y$.

$(-y)=x^2-3x-4$

$-y=x^2-3x-4$ not the same as the original equation when $y$ is replaced by $-y$.

Therefore the graph of the equation is not symmetric with respect to $x\text{-axis}$.

Now test the equation for symmetry with respect to $y\text{-axis}$. Replace $x$ by $-x$.

$y={(-x)}^2-3(-x)-4$

$y=x^2+3x-4$ is not equivalent to the original equation.

Therefore the graph of the equation is not symmetric with respect to $y\text{-axis}$.

Now test the equation for symmetry with respect to the origin. Replace $x$ by $-x$ and $y$ by $-y$.

$(-y)={(-x)}^2-3(-x)-4$

$-y=x^2+3x-4$ is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.