Question

The graph of an equation is shown below.

(a)The x-intercept(s) are$\text{\_\_\_\_}$, and the y-intercept(s) are$\text{\_\_\_\_}$.

(b)The graph is symmetric about the$\text{\_\_\_\_}$(x-axis/y-axis/origin)

Short answer

(a) The x-intercepts are$-3\text{and3}$ and the y-intercepts are $-1\text{and2}$.

(b) The graph is symmetric about y-axis.

Step-by-step solution

a.Step 1. Apply the concept of intercepts

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Step 2. Compute the x-intercept

Consider the graph of the equation.

To find the x-intercept, look for the points whose y-coordinate is 0.

The graph intersects x-axis at two points $-3\text{and 3}$

Therefore, x-intercepts are $-3\text{and 3}$.

Step 3. Compute the y-intercept

To find the y-intercept, look for the points whose x-coordinate is 0.

The graph intersects y-axis at two points $-3\text{and 3}$

Therefore, y-intercepts are $-1\text{and 2}$.

Hence, the x-intercepts are $-3\text{and 3}$ and the y-intercepts are $-1\text{and 2}$.

b.Step 1. Apply the test of symmetry

The function is symmetric about the x-axis, when y is replaced by $-y$, the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by $-x$, the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$, the equation remains unchanged.

Step 2. Test the symmetry about origin

Consider the graph of the equation.

Recall that the function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$, the equation remains unchanged.

From the graph it is observed that if a point of the form $\left(x,y\right)$ lies on the graph then the point $\left(-x,-y\right)$ does not lie on the same. For example $\left(1,2\right)$ lies on the graph of equation but $\left(-1,-2\right)$ does not lie on the graph of the equation.

Therefore, the graph of the equation is not symmetric about the origin.

Step 3. Test the symmetry about axes

Consider the graph of the equation.

Recall that the function is symmetric about the x-axis, when y is replaced by $-y$, the equation remains unchanged.

From the graph it is observed that if a point of the form $\left(x,y\right)$ lies on the graph then the point $\left(x,-y\right)$ does not lie on the same. For example $\left(1,2\right)$ lies on the graph of equation but $\left(1,-2\right)$ does not lie on the graph of the equation.

Therefore, the e the graph of the equation is not symmetric about x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by $-x$, the equation remains unchanged.

From the graph it is observed that if a point of the form $\left(x,y\right)$ lies on the graph then the point $\left(-x,y\right)$ also lie on the same. For example $\left(1,2\right)$ lies on the graph of equation but $\left(-1,2\right)$ also lie on the graph of the equation.

Therefore, the graph of the equation is symmetric about y-axis.