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!
