How do you sketch the graph of #1/(x^2-4)#?

Question

Samantha Adkison · Accepted Answer

Try different x values to get function's value and then plot it

If x is 0, your function #y=-1/4# If x is 1, your function #y=-1/3# If x is 2, your function is not defined. Your function is not defined when #x=-2# too. Plot it accordingly: graph{1/((x^2)-4) [-10, 10, -5, 5]}

Isaiah Allen · Answer

undefined To sketch the graph of $ \frac{1}{x^2 - 4} $, follow these steps:

1. Identify the vertical asymptotes: Set the denominator equal to zero and solve for $ x $. In this case, $ x^2 - 4 = 0 $ gives us $ x = 2 $ and $ x = -2 $. These are the vertical asymptotes.

2. Identify the horizontal asymptote: As $ x $ approaches positive or negative infinity, the function approaches zero. Therefore, the horizontal asymptote is $ y = 0 $.

3. Find the intercepts: The $ y $-intercept occurs when $ x = 0 $, giving $ y = \frac{1}{-4} = -\frac{1}{4} $. There are no $ x $-intercepts.

4. Determine the behavior between asymptotes: Test points in each interval defined by the vertical asymptotes to determine the behavior of the function.

5. Sketch the graph: Using the information gathered from the previous steps, plot the asymptotes, intercepts, and the behavior between asymptotes, and then draw the graph accordingly.

The final graph should show the vertical asymptotes at $ x = 2 $ and $ x = -2 $, a horizontal asymptote at $ y = 0 $, and a curve approaching the asymptotes but never touching or crossing them.