Question

Symmetry and Intercepts (a) Test the equation for symmetry with respect to x axis, the y axis and the origin. (b) Find the x and y intercepts of the graph of the equation.

$y=9-x^2$

Short answer

(a) The graph of equation has symmetry about the y axis..

(b) The value of x intercept is$-3,3$ and the value of y intercept is 9.

Step-by-step solution

Part a. Step 1. Write the given equation.

The given equation is,

$y=9-x^2\mathrm{...}\left(1\right)$

Part a. Step 2. Check for symmetry about the x axis.

Substitute$-y$ for y in equation (1).

$-y=9-x^2\mathrm{...}\left(2\right)$

Equation (2) is not equivalent to equation (1). So the graph is not symmetric about x axis.

Part a. Step 3. Check for symmetry about the y axis.

Substitute$-x$ for x in equation (1).

$$\begin{gathered} y=9-{\left(-x\right)}^2 \\ =9-x^2\mathrm{...}\left(3\right) \end{gathered}$$

Equation (3) is equivalent to equation (1). So the graph is symmetric about y axis.

Thus, the graph of the given equation is symmetric about the y axis.

Part b. Step 1. Write the given equation.

The given equation is,

$y=9-x^2\mathrm{...}\left(4\right)$

Part b. Step 2. Calculate the x intercepts.

Substitute 0 for yin equation (4).

$$\begin{gathered} 0=9-x^2 \\ {x}^2=9 \\ x=\pm 3 \end{gathered}$$

So, the x intercepts are$-3$ and 3.

Part b. Step 3. Calculate the y intercepts.

Substitute 0 for x in equation (4).

$$\begin{gathered} y=9-{\left(0\right)}^2 \\ =9 \end{gathered}$$

So, the y intercept is 9.

Thus, the value of x intercept is$-3,3$ and the value of y intercept is 9.