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.

$6x+y^2=36$

Short answer

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

(b) The value of x intercept is 6 and the value of y intercept is $-6,6$.

Step-by-step solution

Part a. Step 1. Write the given equation.

The given equation is,

$6x+y^2=36\mathrm{...}\left(1\right)$

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

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

$$\begin{gathered} 6x+{\left(-y\right)}^2=36 \\ 6x+y^2=36\mathrm{...}\left(2\right) \end{gathered}$$

Equation (2) is equivalent to equation (1). So the graph is symmetric about x axis.

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

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

$$\begin{gathered} 6\left(-x\right)+y^2=36 \\ -6x+y^2=36\mathrm{...}\left(3\right) \end{gathered}$$

Equation (3) is not equivalent to equation (1). So the graph is not symmetric about y axis.

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

Part b. Step 1. Write the given equation.

The given equation is,

$6x+y^2=36\mathrm{...}\left(4\right)$

Part b. Step 2. Calculate the x intercepts.

Substitute 0 for yin equation (4).

$$\begin{gathered} 6x+{\left(0\right)}^2=36 \\ 6x=36 \\ x=6 \end{gathered}$$

So, the x intercept is 6.

Part b. Step 3. Calculate the y intercepts.

Substitute 0 for x in equation (4).

$$\begin{gathered} 6\left(0\right)+y^2=36 \\ {y}^2=36 \\ y=\pm 6 \end{gathered}$$

So, the y intercept is $-6,6$.

Thus, the value of x intercept is 6 and the value of y intercept is $-6,6$.