Question

In Exercises 71-74. Find the x- and y-intercepts of the graph of the equation.

(a) $y=\sqrt{x^2-16}$

(b) $y=\sqrt{64-x^3}$

Short answer

(a) The x-intercepts are $-4\text{and 4}$ and y-intercept does not exist.

(b) The x-intercept is 4 and y-intercept is 8.

Step-by-step solution

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 equation $y=\sqrt{x^2-16}$.

To find the x-intercept, substitute $y=0$in the equation $y=\sqrt{x^2-16}$.

$\begin{array}{c}0=\sqrt{x^2-16}\\ x^2-16=0\\ x^2=16\\ x=-4,4\end{array}$

Therefore, x-intercepts are $-4\text{and 4}$.

Step 3. Compute the y-intercept

Consider the equation $y=\sqrt{x^2-16}$.

To find the y-intercept, substitute $x=0$in the equation $y=\sqrt{x^2-16}$.

$\begin{array}{c}y=\sqrt{0^2-16}\\ y=\sqrt{-16}\end{array}$

Therefore, no y-intercepts.

Hence, the x-intercepts are $-4\text{and 4}$ and y-intercept does not exist.

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 equation $y=\sqrt{64-x^3}$.

To find the x-intercept, substitute $y=0$in the equation $y=\sqrt{64-x^3}$.

$\begin{array}{c}0=\sqrt{64-x^3}\\ 64-x^3=0\\ x^3=64\\ x=4\end{array}$

Therefore, x-intercept is $4$.

Step 3. Compute the y-intercept

Consider the equation $y=\sqrt{64-x^3}$.

To find the y-intercept, substitute $x=0$in the equation $y=\sqrt{64-x^3}$.

$\begin{array}{c}y=\sqrt{64-0^3}\\ y=\sqrt{64}\\ y=8\end{array}$

Therefore, y-intercept is $8$.

Hence, the x-intercept is 4 and y-intercept is 8.