Explanation
Finding the x-intercepts
To find the x-intercepts, we substitute y = 0 and solve for x.
[tex]\begin{gathered} y=\frac{x^2-25}{x^2-16} \\ 0=\frac{x^{2}-25}{x^{2}-16} \end{gathered}[/tex]
The equation is equal to 0 if the numerator is equal to 0. Then, we solve the numerator.
[tex]\begin{gathered} 0=x^2-25 \\ \text{ Add 25 from both sides} \\ 0+25=x^2-25+25 \\ 25=x^2 \\ $\text{ Apply square root to both sides of the equation}$ \\ \sqrt{25}=\sqrt{x^2} \\ \pm5=x \\ \text{ Then} \\ x=5,-5 \end{gathered}[/tex]
Thus, the x-intercepts of the equation are the ordered pairs (5,0) and (-5,0).
Finding the y-intercepts
To find the y-intercepts, we substitute x = 0 and solve for y.
[tex]\begin{gathered} y=\frac{x^{2}-25}{x^{2}-16} \\ y=\frac{0^2-25}{0^2-16} \\ y=\frac{0^-25}{0-16} \\ y=\frac{25}{16} \end{gathered}[/tex]
Thus, the y-intercept of the equation is the ordered pair (0,25/16).
Answer
The intercepts of the graph of the given equation are the ordered pairs (5,0), (-5,0), and (0,25/16).
