F(x)= x^2 - 4x / x^2 - x - 2 find the x and y intercepts

find vertical and horizontal asymptotes

determine the holes
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Step-by-step explanation:

First, I decided to determine the vertical asymptote! You can determine the vertical asymptote by taking the denominator and setting it equal to zero.

−4x+4=0

−4x=−4

x=1

There's a vertical asymptote at x=1.

Now for the horizontal asymptote! The horizontal asymptote is determined by looking at the leading coefficients in the numerator and denominator. Here's a helpful visual.

content://com.cryptotab.android.FileProvider/images/screenshot/16159519510813476709092178134526.jp

In our case, the leading coefficient in the numerator (x2) is larger than the leading coefficient in the denominator (−4x). This means there is no horizontal asymptote!

In some cases, a slant asymptote may be present in a graph. These are sometimes present in graphs that have a stronger leading coefficient in the numerator than the denominator. To determine the slant asymptote, we need to divide the numerator by the denominator.

content://com.cryptotab.android.FileProvider/images/screenshot/16159520052041557972364765603410

The slant asymptote is y=−x4−1

Holes are determined by finding numbers that when plugged into x, return undefined. On a graphing calculator these values will give us an error. Luckily for us, there are no holes, so we can move on.

X-intercepts are values that make the numerator equal zero. We can find these by setting the numerator equal to zero and solving.

x2+3x=0

x(x+3)=0

x+3=0

x=−3

Another number that makes the numerator equal zero is, well, zero!

x2+3x=0

(0)2+3(0)=0

0=0

So we know our x-intercepts are 0 and -3.

To find the y-intercepts, set x equal to zero and solve the equation!

y=x2+3x−4x+4

y=(0)2+3(0)−4(0)+4

y=04

y=0

So we have a y-intercept at y=0!