Chapter 1.2, Problem 36AYU

In Problems 33-44, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, or the origin.

To determine

To find: The intercepts and to indicate whether the graph is symmetric with respect to the $x\text{-axis}$, $y\text{-axis}$ and the origin.

Answer to Problem 36AYU

The $y\text{-intercept}$ is $\left(0,1\right)$. The graph is symmetric with respect to $y\text{-axis}$.

Explanation of Solution

Given:

The graph of an equation

Calculation:

From the given graph, we can identify that the intercepts are $\left(-2,0\right)$, $\left(0,3\right)$ and $\left(2,0\right)$. The $x\text{-intercepts}$ are $(-2,0)$ and $\left(2,0\right)$. The $y\text{-intercept}$ is $\left(0,3\right)$. The $x\text{-intercepts}$ have $y$ coordinates equal to 0 and the $y\text{-intercepts}$ have $x$ coordinates equal to 0

If a graph is symmetric with respect to the $x\text{-axis}$ and the point $\left(-x,y\right)$ is on the graph then the point $\left(-x,-y\right)$ is also on the graph. Also the point $\left(x,y\right)$ is on the graph then the point $\left(x,-y\right)$ is also on the graph. But here in this graph both the points are not on the graph. Hence the graph is not symmetric with respect to $\text{x-axis}$.

If a graph is symmetric with respect to the $y\text{-axis}$ and the point $\left(-2,0\right)$ is on the graph then the point $\left(2,0\right)$ is also on the graph.

If a graph is symmetric with respect to the origin and the point $\left(x,y\right)$ is on the graph then the point $\left(-x,-y\right)$ is on the graph. Also the point $\left(-x,y\right)$ is on the graph then the point $\left(x,-y\right)$ is also on the graph. But this graph is not symmetric with respect to the origin.