Question

An equation and its graph are given. Find the x- and y-intercepts.

$x^2+y^3-x^2{y}^2=64$

Short answer

The x-intercepts are 8 and -8, they-intercept is 4.

Step-by-step solution

Step 1. Apply the concept of intercepts.

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

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

Step 2. Compute the x-intercept.

Consider the provided equation and the graph of the equation is sketched below,

Observe from the graph that the curve intersects thex-axis at two points and 8.

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

Substitute $y=0$in the equation $x^2+y^3-x^2{y}^2=64$.

$$\begin{gathered} x^2+{\left(0\right)}^3-x^2{\left(0\right)}^2=64 \\ {x}^2=64 \end{gathered}$$

Either $x=-8$or $x=8$.

Therefore, x-intercepts are $-8$and 8.

Step 3. Compute the y-intercept.

Consider the provided equation$x^2+y^3-x^2{y}^2=64$and the graph of the equation is sketched below.

Observe from the graph that the curve intersects they-axis at one point 4.

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

Substitute $x=0$in the equation $x^2+y^3-x^2{y}^2=64$,

$$\begin{gathered} 0^2+y^3-x^2{\left(0\right)}^2=64 \\ {y}^3=64 \\ y=4 \end{gathered}$$

Therefore, they-intercept is 4.

Thus, the x-intercepts are$-8$ and 8 and they-intercept is 4.