Find the intercepts of the graph of the equation $y=\frac{x^2-4}{x^2-9}$.

The y-intercept of the graph is 9 4 ​ , and the x-intercepts are 2 and − 2 . To find the y-intercept, set x = 0 and for x-intercepts, set y = 0 and solve the numerator equation. Both intercepts have been verified to ensure they do not make the denominator zero.
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Answered by Anonymous | 2025-07-04

To find the y-intercept, set x = 0 and solve for y , which gives y = 0 2 − 9 0 2 − 4 ​ = 9 4 ​ .
To find the x-intercepts, set y = 0 and solve for x . This means solving x 2 − 4 = 0 , which gives x = ± 2 .
Check that the solutions x = 2 and x = − 2 do not make the denominator x 2 − 9 equal to zero. Since 2 2 − 9 = − 5  = 0 and ( − 2 ) 2 − 9 = − 5  = 0 , both are valid x-intercepts.
The y-intercept is 9 4 ​ and the x-intercepts are 2 and − 2 , so the final answer is: x = 2 , x = − 2 , y = 9 4 ​ ​

Explanation

Understanding the Problem We are given the equation y = x 2 − 9 x 2 − 4 ​ and we want to find the intercepts of its graph. The intercepts are the points where the graph intersects the x-axis and the y-axis.

Finding the y-intercept To find the y-intercept, we set x = 0 and solve for y . Substituting x = 0 into the equation gives us y = 0 2 − 9 0 2 − 4 ​ = − 9 − 4 ​ = 9 4 ​ So the y-intercept is 9 4 ​ .

Finding the x-intercepts To find the x-intercepts, we set y = 0 and solve for x . This gives us the equation x 2 − 9 x 2 − 4 ​ = 0 A fraction is equal to zero if and only if its numerator is equal to zero and its denominator is not equal to zero. Thus, we need to solve x 2 − 4 = 0 for x , and make sure that x 2 − 9  = 0 .

Solving for x-intercepts Solving x 2 − 4 = 0 gives us x 2 = 4 , so x = ± 4 ​ = ± 2 . Thus, x = 2 or x = − 2 . We need to check that these values do not make the denominator equal to zero. If x = 2 , then x 2 − 9 = 2 2 − 9 = 4 − 9 = − 5  = 0 .
If x = − 2 , then x 2 − 9 = ( − 2 ) 2 − 9 = 4 − 9 = − 5  = 0 .
Since neither value makes the denominator zero, the x-intercepts are x = 2 and x = − 2 .

Final Answer Therefore, the y-intercept is 9 4 ​ , and the x-intercepts are 2 and − 2 .


Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, the x-intercept of a cost function can represent the break-even point, where costs equal revenue. In physics, intercepts on a graph of motion can indicate starting points or points of rest. Knowing how to find intercepts helps in analyzing data and making informed decisions in many fields.

Answered by GinnyAnswer | 2025-07-04