Question

Find the intercepts of the graph of the equation \(y=\frac{x^{2}-1}{x^{2}-4}\)

Short answer

Y-intercept at (0, \( \frac{1}{4} \)). X-intercepts at (1, 0) and (-1, 0).

Step-by-step solution

- Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. \[ y = \frac{0^2 - 1}{0^2 - 4} = \frac{-1}{-4} = \frac{1}{4} \] So, the y-intercept is at (0, \( \frac{1}{4} \)).

- Find the x-intercepts

To find the x-intercepts, set y to 0 in the equation and solve for x. \[ 0 = \frac{x^2 - 1}{x^2 - 4} \]The fraction is zero when the numerator is zero (and the denominator is not zero). So, \[ x^2 - 1 = 0 \]Solve for x: \[ x^2 = 1 \]\[ x = \pm 1 \]Therefore, the x-intercepts are at (1, 0) and (-1, 0).

Key concepts

y-intercept

To find the y-intercept of a rational function, we need to set the value of x to 0 and solve for y. This is because the y-intercept is the point where the graph crosses the y-axis, and at this point, x is always zero.
In the given equation \[ y = \frac{x^2 - 1}{x^2 - 4} \]we will set x to 0:
\[ y = \frac{0^2 - 1}{0^2 - 4} = \frac{-1}{-4} = \frac{1}{4} \]
So, the y-intercept is at (0, \( \frac{1}{4} \)).
Finding the y-intercept helps us understand a specific point where the graph touches the y-axis, giving us basic insight into the behavior of the function.

x-intercept

To find the x-intercept of a rational function, we need to set y to 0 and solve for x. This is because the x-intercept is where the graph crosses the x-axis, and at these points, y is always zero.
Given the equation \[ y = \frac{x^2 - 1}{x^2 - 4} \]we set y to 0:
\[ 0 = \frac{x^2 - 1}{x^2 - 4} \]
A fraction equals zero when its numerator equals zero, as long as the denominator is not zero. Therefore, we set the numerator of the fraction to zero:
\[ x^2 - 1 = 0 \]
We solve for x:
\[ x^2 = 1 \]\[ x = \pm 1 \]
Thus, the x-intercepts are at (1, 0) and (-1, 0). Finding the x-intercepts provides critical points where the function crosses the x-axis, showing us key values where the output of the function is zero.

solving equations

Solving equations is a fundamental skill in mathematics, especially when finding intercepts of rational functions. Different techniques are used depending on the type of function and the goal. Here’s a simple approach to solving the equations for intercepts:

• **For the y-intercept**, substitute x with 0 in the equation and solve for y.
• **For the x-intercept**, set the entire equation equal to 0. Then solve for x by simplifying and isolating the variable.

When dealing with the rational function \[ y = \frac{x^2 - 1}{x^2 - 4} \]to find the y-intercept, setting x to 0 is straightforward. Calculations show:
\[ y = \frac{-1}{-4} = \frac{1}{4} \]For the x-intercept, setting y to 0 leads us to isolate x by setting the numerator to zero:
\[ x^2 - 1 = 0 \]Factoring or applying square roots gives us the solutions:
\[ x = \pm 1 \]Exploring solving equations aids in understanding their structure and methodically applying logic to find meaningful solutions.