Section 4.8 : Rational Functions
1. Sketch the graph of the following function. Clearly identify all intercepts and asymptotes.
f(x)=−4x−2Show All Steps Hide All Steps
Start SolutionLet’s first find the intercepts for this function.
The y-intercept is the point (0,f(0))=(0,2).
For the x-intercepts we set the numerator equal to zero and solve. However, in this case the numerator is a constant (-4 specifically) and so can’t ever be zero. Therefore, this function will have no x-intercepts.
Show Step 2We can find any vertical asymptotes be setting the denominator equal to zero and solving. Doing that for this function gives,
x−2=0→x=2So, we’ll have a vertical asymptote at x=2.
Show Step 3For this equation the largest exponent of x in the numerator is zero since the numerator is a constant. The largest exponent of x in the denominator is 1, which is larger than the largest exponent in the numerator, and so the x-axis will be the horizontal asymptote.
Show Step 4From Step 2 we saw we only have one vertical asymptote and so we only have two regions to our graph : x<2 and x>2.
We’ll need a point in each region to determine if it will be above or below the horizontal asymptote. Here are a couple of function evaluations for the points.
f(0)=2→(0,2)f(3)=−4→(3,−4)Note that the first evaluation didn’t really need to be done since it was just the y-intercept which we had already found in the first step. It was included here mostly for the sake of completeness.
Show Step 5Here is a sketch of the function with the points found above. The vertical asymptote is indicated with a blue dashed line and recall that the horizontal asymptote is just the x-axis.
