Graphs of Rational Functions

Graphs of functions with x in the denominator of a fraction

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Graphs of Rational Functions
We have investigated rational functions in prior lessons, and described them loosely as functions that involve the division of polynomials.

Can the answers to a rational function be found by graphing? How would you go about graphing them?

Graphing Rational Functions

Any function that has the form

\begin{align*}f(x)=\frac{P(x)}{Q(x)}\end{align*}

where \begin{align*}P(x)\end{align*} and \begin{align*}Q(x)\end{align*} are polynomials and \begin{align*}Q(x)\ne0\end{align*}, is called a rational function. The domain of any rational function includes all real numbers \begin{align*}x\end{align*} that do not make the denominator zero.

Just like polynomials, rational functions can be graphed using transformations. The main point to remember for graphing rational functions by transformations is that some transformations change the asymptotes while others do not.

• \begin{align*}r(x)+c\end{align*} is a vertical shift which moves each horizontal asymptote up \begin{align*}c\end{align*} units (or down if \begin{align*}c<0\end{align*}).
• \begin{align*}r(x-c)\end{align*} is a horizontal shift which moves each vertical asymptote right \begin{align*}c\end{align*} units (or left if \begin{align*}c<0\end{align*}).
• \begin{align*}a\cdot r(x)\end{align*} is a vertical stretch which moves horizontal asymptotes by a multiple of \begin{align*}a\end{align*} (so this moves the horizontal asymptote closer to the \begin{align*}x-\end{align*}axis if \begin{align*}a<0\end{align*}.
• \begin{align*}r(a\cdot x)\end{align*} is a horizontal compression which moves the vertical asymptotes closer to \begin{align*}y-\end{align*}axis by a factor of \begin{align*}\frac{1}{a}\end{align*}.
• \begin{align*}r(-x)\end{align*} is a reflection about the \begin{align*}y-\end{align*}axis. All vertical asymptotes are also reflected.
• \begin{align*}-r(x)\end{align*} is a reflection about the \begin{align*}x-\end{align*}axis. All horizontal asymptotes are also reflected.

Examples

Example 1

What is the domain of \begin{align*}f(x)=\frac{1}{x}\end{align*}?

Notice that the only input that can make the denominator equal to zero is \begin{align*}x=0\end{align*}. Thus we say that the domain of \begin{align*}f(x)\end{align*} is all real numbers except \begin{align*}x=0\end{align*}. When looking at the graph of \begin{align*}f(x)=\frac{1}{x}\end{align*} (see Example 2 below), you will notice that as \begin{align*}x\end{align*} approaches 0 from the left, \begin{align*}f(x)\end{align*} decreases. But when \begin{align*}x\end{align*} approaches 0 from the right, \begin{align*}f(x)\end{align*} increases. Because of this behavior, the \begin{align*}x-\end{align*}axis and \begin{align*}y-\end{align*}axis play the role of horizontal and vertical asymptotes, respectively.

Example 2

Graph the function \begin{align*}f(x)=\frac{1}{x}\end{align*}.

We know that the domain of \begin{align*}f(x)\end{align*} is all real numbers excluding \begin{align*}x=0\end{align*}. The vertical line \begin{align*}x=0\end{align*} is called a vertical asymptote. For \begin{align*}x<0, f(x)<0,\end{align*} and for \begin{align*}x>0, f(x)>0\end{align*}. Plotting a few sample points should indicate the shape of \begin{align*}f(x)\end{align*}.

\begin{align*}& x && 1 && 2 && 10 && \frac{1}{5} && \frac{1}{10} && -1 && -\frac{1}{2} && -\frac{1}{10} && -2 && -10\\ & f(x)=\frac{1}{x} && 1 && \frac{1}{2} && \frac{1}{10} && 5 && 10 && -1 && -2 && -10 && -\frac{1}{2} && -\frac{1}{10}\end{align*}

Example 3

A rational function \begin{align*}r(x)\end{align*} is shown in the figure below. Use the graph of \begin{align*}r(x)\end{align*} to sketch graphs of:

a. \begin{align*}r(x)-3\end{align*}

The horizontal asymptote moves down by three units.

b. \begin{align*}-r(x)\end{align*}

The function is reflected about the \begin{align*}x-\end{align*}axis so the horizontal asymptote is also reflected.

c. \begin{align*}r(3-x)\end{align*}

\begin{align*}r(3-x) = r(-(x-3))\end{align*}. First, graph \begin{align*}r(-x)\end{align*}, and then shift that graph three units to the right to get \begin{align*}r(-(x-3))\end{align*}. The new vertical asymptote is \begin{align*}x = 1\end{align*}.

Example 4

For the following rational function, determine the domain, the asymptotes, the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}intercepts and then sketch the graph.

\begin{align*}f(x)=\frac{x+2}{x^{2}+1}\end{align*}

To identify domain limitations, find value(s) which make the denominator = 0: In this case, where \begin{align*}x^2\end{align*}, the only variable in the denominator, is added to 0, any value for x will be positive. So the domain is all real numbers.

With no limitations on the domain, there are no vertical asymptotes.

The horizontal asymptote: \begin{align*}y=0\end{align*} becomes apparent as x becomes huge and the "+2" and "+1" no longer have an effect, giving: \begin{align*}f(x) = \frac{x}{x^2} \to f(x) = \frac{1}{x} \to f(x) = 0\end{align*} So the horizontal asymptote is 0

Example 5

For the following rational function, determine the domain, the asymptotes, the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}intercepts and then sketch the graph.

\begin{align*}f(x)=\frac{9x^{2}-4}{3x+2}\end{align*}

For this problem, the first step is to recognize the difference of squares in the numerator: \begin{align*}9x^2 - 4\end{align*} which factors easily into \begin{align*}(3x - 2)(3x + 2)\end{align*}

This gives: \begin{align*}f(x) = \frac{(3x - 2)(3x + 2)}{(3x + 2)}\end{align*}

NOTE: Although we can cancel the (3x + 2) to evaluate the general behavior of the function, giving: \begin{align*}f(x) = 3x + 2\end{align*} (which is just a line), that does not change the fact that a zero in the denominator of the equation is undefined, it just means that the rest of the function behaves like \begin{align*}y = 3x - 2\end{align*}.

To identify domain limitations, find value(s) which make the denominator = 0: In this case: \begin{align*}3x + 2 = 0 \to 3x = -2 \to x = -\frac{2}{3}\end{align*} which means the domain cannot include a value of -2/3 for x: \begin{align*}x\ne-\frac{2}{3}\end{align*}

Since the equation behaves as a line with a single non-existent value, there is no vertical asymptote.

As x gets huge, we end up with \begin{align*}f(x) = \frac{9x^2}{3x} \to f(x) = 3x\end{align*} which is a slant asymptote that parallels the line of the function!

Review

1. What is the hole in the graph of a function called?
2. How are polynomial graphs different from rational function graphs?
3. What does the graph of the simplified function that is continuous everywhere not have?
4. Why can't rational function be simplified to determine the domain?
5. Put the following steps into the correct order for graphing rational functions. a) Find all asymptotes. b) Sketch a smooth graph based on the information. c) Factor numerator and denominator completely and put in lowest terms. Identify any holes. d) Determine the behavior around the vertical asymptotes using a table of signs. e) Find all intercepts. f) Find the places where the function crosses the horizontal asymptote/oblique asymptote.
6. Simplify \begin{align*}\frac{6x^2 + 21x + 9}{4x^2 - 1}\end{align*}
7. Given: \begin{align*}y = \frac{x^2 + 2x - 8}{x}\end{align*}. Find the intercepts, any asymptotes, and identify end behavior.
8. Graph the equation and state the domain.

Simplify each function in questions 9 and 10 and state any value(s) of x that make the function undefined.

1. \begin{align*}f(x)=\frac{3 - x}{x^2 - 3x}\end{align*}
2. \begin{align*}f(x) = \frac{x^2 - 2x - 15}{x - 5}\end{align*}

For each function in questions 11-13, identify points of discontinuity and the form of the graph of the simplified function (linear or quadratic).

1. \begin{align*}f(x) = \frac{ 2x^2 + x - 1}{2x - 1}\end{align*}
2. \begin{align*}f(x) = \frac{-2x^3 + 9x^2 - 10x + 3}{x - 3}\end{align*}
3. \begin{align*}f(x) = \frac{x^3 - 13x -12}{x^2 - 3x - 4}\end{align*}
4. Write a quadratic function where the domain cannot include 5 or -5, and the graph has two asymptotes
5. Simplify, identify the asymptotes and intercepts, and sketch the graph of \begin{align*}f(x) = \frac{x^3 - 64x}{x^2 - 16}\end{align*}

To see the Review answers, open this PDF file and look for section 2.7.

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Asymptotes An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).
compression A stretch or compression is a function transformation that makes a graph narrower or wider, without translating it horizontally or vertically.
domain The domain of a function is the set of $x$-values for which the function is defined.
Function A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Horizontal Asymptote A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.
Oblique Asymptote An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.
Polynomial Function A polynomial function is a function defined by an expression with at least one algebraic term.
Reflection Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.
shift A shift, also known as a translation or a slide, is a transformation applied to the graph of a function that does not change the shape or orientation of the graph, only the location of the graph.
shifts A shift, also known as a translation or a slide, is a transformation applied to the graph of a function that does not change the shape or orientation of the graph, only the location of the graph.
Slant Asymptote A slant asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but will never reach. A slant asymptote exists when the numerator of the function is exactly one degree greater than the denominator. A slant asymptote may be found through long division.
stretch A stretch or compression is a function transformation that makes a graph narrower or wider.
Transformations Transformations are used to change the graph of a parent function into the graph of a more complex function.
Vertical Asymptote A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.