Question

Sketch the graph ofid="TexFormula1" title="f(x) = \frac{x - 3}{x + 4 } " alt="f(x) = \frac{x - 3}{x + 4 } " align="absmiddle" class="latex-formula">

Answer 1

First find any vertical asymptotes:
Any x-values which make the denominator equal 0
---> x+4 = 0   ----->  x = -4
Draw a vertical dotted line along x = -4

Next find any horizontal asymptotes:
A y-value that function approaches as x gets very, very large
Plug in x = 1000, then x = 1000,000. You will see the y-value get very,very close to 1.
In general if the exponents of top/bottom are equal, then the horizontal asymptote is the ratio of the leading coefficients.
Draw a horizontal dotted line along y= 1

Next pick 4 x-values, two on either side of the vertical asymptote.
x = 0 , x = -2, x = -8, x = -6
Plot the 4 points ----> (0,-3/4) , (-2,-5/2),  (-6,9/2) , (-8,11/4)

Notice 2 points will be separated from the other 2 by the dotted lines.
For each pair of points, Draw a curved line through them so they approach but NOT cross the dotted lines.