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Section 11.1 Graphing Rational Functions
Subsection 11.1.1 Sketching the Graph of a Rational Function Summary
We can use all the previous properties of rational functions in order to graph them. Then,
Factor and simplify .
If possible, factor the numerator and denominator.
Determine the domain of the function, which is all real numbers, except the roots of the denominator.
Simplify by cancelling any common factors.
Find any holes . Cancelled factors of the form \(x - c\) will result in a hole at \(x = c\text{,}\) and the \(y\) -coordinate of these holes will come from evaluating the simplified function at \(x = c\text{.}\)
Find properties of the function:
Find the y-intercept (if it exists) . The \(y\) -intercept comes from substituting \(x = 0\) (i.e. finding \(f(0)\) ) (as long as 0 is in the domain of \(f\) ).
Find any x-intercepts . Any \(x\) -intercepts of \(f\) are zeros of the numerator.
Find any vertical asymptotes . Any vertical asymptotes are from the zeros of the denominator.
Find the horizontal asymptote (or oblique asymptote) . Compare the degree of the numerator and the denominator.
If \(\text{num} \lt \text{denom}\text{,}\) then the line \(y = 0\) (the \(x\) -axis) is the horizontal asymptote.
If \(\text{num} = \text{denom}\text{,}\) then the line \(y = \frac{\text{leading coefficient of num}}{\text{leading coefficient of denom}}\text{,}\) the ratio of the leading coefficients, is the horizontal asymptote.
If \(\text{num} \gt \text{denom}\text{,}\) then there is no horizontal asymptote.
Plot additional points , if necessary. A good guideline is to have at least 1 point on each side of each vertical asymptote. Two points can be better if you need more detail.
Sketch the graph by connecting the points. The graph should get closer and closer to each asymptote. Be sure to not cross any vertical asymptote.
Subsection 11.1.2 Examples
Exercise Group 11.1.1 . Graphing Rational Functions 1.
Sketch the graph of each function. State the equations of any asymptotes, the
\(x\) -intercept(s), the
\(y\) -intercept, and the coordinates of any holes.
(a) \(f(x)=\dfrac{x+1}{x^2-2x-8}\) Answer .
VA:
\(x=-2,4\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((-1,0)\text{,}\) \(y\) -int:
\((0,-\frac{1}{8})\)
(b) \(f(x)=\dfrac{x-4}{x^2-4}\) Answer .
VA:
\(x=-2,2\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((4,0)\text{,}\) \(y\) -int:
\((0,1)\)
(c) \(f(x)=\dfrac{x^2-4}{x^2-9}\) Answer .
VA:
\(x=-3,3\text{,}\) HA:
\(y=1\text{,}\) \(x\) -int:
\((-2,0),(2,0)\text{,}\) \(y\) -int:
\((0,\frac{4}{9})\)
(d) \(f(x)=\dfrac{x+3}{x^2+7x+12}\) Answer .
VA:
\(x=-4\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int: none,
\(y\) -int:
\((0,\frac{1}{4})\text{,}\) hole:
\((-3,1)\)
(e) \(f(x)=\dfrac{x^2+x-2}{x+1}\) Answer .
VA:
\(x=-1\text{,}\) HA: none,
\(x\) -int:
\((-2,0),(1,0)\text{,}\) \(y\) -int:
\((0,-2)\)
(f) \(f(x)=\dfrac{3x^2+8x+4}{x^2-x-2}\) Answer .
VA:
\(x=-1,2\text{,}\) HA:
\(y=3\text{,}\) \(x\) -int:
\((-2,0),(-\frac{2}{3},0)\text{,}\) \(y\) -int:
\((0,-2)\)
(g) \(f(x)=\dfrac{x}{x^2-3x-4}\) Answer .
VA:
\(x=-1,4\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((0,0)\text{,}\) \(y\) -int:
\((0,0)\)
(h) \(f(x)=\dfrac{x-2}{x^2+5x+6}\) Answer .
VA:
\(x=-3,-2\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((2,0)\text{,}\) \(y\) -int:
\((0,-\frac{1}{3})\)
(i) \(f(x)=\dfrac{x^2-3x+2}{1-x^2}\) Answer .
VA:
\(x=-1\text{,}\) HA:
\(y=-1\text{,}\) \(x\) -int:
\((2,0)\text{,}\) \(y\) -int:
\((0,2)\text{,}\) hole:
\((1,\frac{1}{2})\)
(j) \(f(x)=\dfrac{2x^2-5x-3}{x^2-5x+6}\) Answer .
VA:
\(x=2\text{,}\) HA:
\(y=2\text{,}\) \(x\) -int:
\((-\frac{1}{2},0)\text{,}\) \(y\) -int:
\((0,-\frac{1}{2})\text{,}\) hole:
\((3,7)\)
(k) \(f(x)=\dfrac{-x}{x^2-9}\) Answer .
VA:
\(x=-3,3\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((0,0)\text{,}\) \(y\) -int:
\((0,0)\)
(l) \(f(x)=\dfrac{-3x^2+11x-6}{x^2-x-6}\) Answer .
VA:
\(x=-2\text{,}\) HA:
\(y=-3\text{,}\) \(x\) -int:
\((\frac{2}{3},0)\text{,}\) \(y\) -int:
\((0,1)\text{,}\) hole:
\((3,-\frac{7}{5})\)
(m) \(f(x)=\dfrac{2x^2-8}{x-2}\) Answer .
VA: none, HA: none,
\(x\) -int:
\((-2,0)\text{,}\) \(y\) -int:
\((0,4)\text{,}\) hole:
\((2,8)\)
Exercise Group 11.1.2 . Graphing Rational Functions 2.
Sketch the graph of each function. State the equations of any asymptotes, the
\(x\) -intercept(s), the
\(y\) -intercept, and the coordinates of any holes.
(a) \(f(x)=\dfrac{2x^2-3x+1}{x-2}\) Answer .
VA:
\(x=2\text{,}\) SA:
\(y=2x+1\text{,}\) \(x\) -int:
\((\frac{1}{2},0),(1,0)\text{,}\) \(y\) -int:
\((0,-\frac{1}{2})\)
(b) \(f(x)=\dfrac{x^2+2x-15}{x^2-4x+3}\) Answer .
VA:
\(x=1\text{,}\) HA:
\(y=1\text{,}\) \(x\) -int:
\((-5,0)\text{,}\) \(y\) -int:
\((0,-5)\text{,}\) hole:
\((3,4)\)
(c) \(f(x)=\dfrac{x+2}{x^2+3x+2}\) Answer .
VA:
\(x=-1\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int: none,
\(y\) -int:
\((0,1)\text{,}\) hole:
\((-2,-1)\)
(d) \(f(x)=\dfrac{x^2+5x+6}{x^2-4x-21}\) Answer .
VA:
\(x=7\text{,}\) HA:
\(y=1\text{,}\) \(x\) -int:
\((-2,0)\text{,}\) \(y\) -int:
\((0,-\frac{2}{7})\text{,}\) hole:
\((-3,\frac{1}{10})\)
(e) \(f(x)=\dfrac{x^2+5x+4}{x+1}\) Answer .
VA: none, HA: none,
\(x\) -int:
\((-4,0)\text{,}\) \(y\) -int:
\((0,4)\text{,}\) hole:
\((-1,3)\)
(f) \(f(x)=\dfrac{x^2+5x-14}{x^2-6x+8}\) Answer .
VA:
\(x=4\text{,}\) HA:
\(y=1\text{,}\) \(x\) -int:
\((-7,0)\text{,}\) \(y\) -int:
\((0,-\frac{7}{4})\text{,}\) hole:
\((2,-\frac{9}{2})\)
(g) \(f(x)=\dfrac{x^2+5x}{x^2+7x+10}\) Answer .
VA:
\(x=-2\text{,}\) HA:
\(y=1\text{,}\) \(x\) -int:
\((0,0)\text{,}\) \(y\) -int:
\((0,0)\text{,}\) hole:
\((-5,\frac{5}{3})\)
(h) \(f(x)=\dfrac{x^2-7x+12}{x^2-9}\) Answer .
VA:
\(x=-3\text{,}\) HA:
\(y=1\text{,}\) \(x\) -int:
\((4,0)\text{,}\) \(y\) -int:
\((0,-\frac{4}{3})\text{,}\) hole:
\((3,-\frac{1}{6})\)
(i) \(f(x)=\dfrac{2x^2+5x-3}{x+3}\) Answer .
VA: none, HA: none,
\(x\) -int:
\((\frac{1}{2},0)\text{,}\) \(y\) -int:
\((0,-1)\text{,}\) hole:
\((-3,-7)\)
(j) \(f(x)=\dfrac{x+4}{x^2-3x-4}\) Answer .
VA:
\(x=-1,4\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((-4,0)\text{,}\) \(y\) -int:
\((0,-1)\)
(k) \(f(x)=\dfrac{x+4}{x^2+5x+4}\) Answer .
VA:
\(x=-1\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int: none,
\(y\) -int:
\((0,1)\text{,}\) hole:
\((-4,-\frac{1}{3})\)
(l) \(f(x)=\dfrac{x^2+4x}{x+4}\) Answer .
VA: none, HA: none,
\(x\) -int:
\((0,0)\text{,}\) \(y\) -int:
\((0,0)\text{,}\) hole:
\((-4,-4)\)
(m) \(f(x)=\dfrac{2x^2+7x-15}{9-4x^2}\) Answer .
VA:
\(x=-\frac{3}{2}\text{,}\) HA:
\(y=-\frac{1}{2}\text{,}\) \(x\) -int:
\((-5,0)\text{,}\) \(y\) -int:
\((0,-\frac{5}{3})\text{,}\) hole:
\((\frac{3}{2},-\frac{13}{12})\)
Exercise Group 11.1.3 . Advanced Examples.
Sketch the graph of each function. State the equations of any asymptotes, the
\(x\) -intercept(s), the
\(y\) -intercept, and the coordinates of any holes.
(a) \(f(x)=\dfrac{x^3-4x^2+x+6}{x^2 + 3x + 2}\)
Hint .
Factor the cubic using synthetic division.
Answer .
VA:
\(x=-2\text{,}\) SA:
\(y=x-7\text{,}\) \(x\) -int:
\((2,0),(3,0)\text{,}\) \(y\) -int:
\((0,3)\text{,}\) hole:
\((-1,12)\)
(b) \(f(x)=\dfrac{x^3+3x^2-6x-8}{x^2 + 3x - 4}\)
Hint .
Factor the cubic using factor by grouping (or synthetic division).
Answer .
VA:
\(x=1\text{,}\) SA:
\(y=x\text{,}\) \(x\) -int:
\((-1,0),(2,0)\text{,}\) \(y\) -int:
\((0,2)\text{,}\) hole:
\((-4,-\frac{18}{5})\)
(c) \(f(x)=\dfrac{x^3+6x^2+9x+4}{x^2+x-2}\)
Hint .
Factor the cubic using synthetic division.
Answer .
VA:
\(x=-2,1\text{,}\) SA:
\(y=x+5\text{,}\) \(x\) -int:
\((-4,0),(-1,0)\text{,}\) \(y\) -int:
\((0,-2)\text{,}\) hole: none
(d) \(f(x)=\dfrac{x^2}{x^3 - 3x^2 - x + 3}\)
Hint .
Factor the denominator using factor by grouping (or synthetic division).
Answer .
VA:
\(x=-1,1,3\text{,}\) HA:
\(y=0\text{,}\) \(x\) -int:
\((0,0)\text{,}\) \(y\) -int:
\((0,0)\text{,}\) hole: none
(e) \(f(x)=\dfrac{x^4-2x^3-7x^2+8x+12}{x^3-7x+6}\) Answer .
VA:
\(x=-3,1\text{,}\) SA:
\(y=x-2\text{,}\) \(x\) -int:
\((-2,0),(-1,0),(3,0)\text{,}\) \(y\) -int:
\((0,2)\text{,}\) hole:
\((2,-\frac{12}{5})\)
