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# Graphs of Rational Functions

## Graphs of functions with x in the denominator of a fraction

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Graphs of Rational Functions

Suppose Susan and Victor both graph the function y=85x\begin{align*}y= \frac{8}{5-x}\end{align*} on their graphing calculators. Susan says that the graph looks incorrect because there should be a hole in the graph at x=5\begin{align*}x=5\end{align*}. Victor says that the graph looks fine and that the graphing calculator would never make a mistake. Who do you think is right? Why do you think this?

### Graphing Rational Functions

Previously, you learned the basics of graphing an inverse variation function. The hyperbola forms two branches in opposite quadrants. The axes are asymptotes to the graph. This section  will compare graphs of inverse variation functions. You will also learn how to graph other rational equations.

#### Graphing Functions for Given Values

Let's graph the function f(x)=kx\begin{align*}f(x)=\frac{k}{x}\end{align*} for the following values of k\begin{align*}k\end{align*}:

k=2,1,12,1,2,4\begin{align*}k=-2, -1, -\frac{1}{2}, 1, 2, 4\end{align*}

The graphs for the values k=1,k=2,\begin{align*}k = 1, k = 2,\end{align*} and k=(12)\begin{align*}k = \left(\frac{1}{2} \right )\end{align*} in Quadrant I are shown. Try your hand at filling in Quadrant III for these graphs as well as graphing the other values.

Remember, as mentioned in the previous section, if k\begin{align*}k\end{align*} is positive, then the branches of the hyperbola are located in quadrants I and III. If k\begin{align*}k\end{align*} is negative, the branches are located in quadrants II and IV. Also notice how the hyperbola changes as k\begin{align*}k\end{align*} gets larger.

#### Rational Functions and Asymptotes

A rational function is a ratio of two polynomials (a polynomial divided by another polynomial). The formal definition is:

f(x)=g(x)h(x),where h(x)0\begin{align*}f(x)=\frac{g(x)}{h(x)}, \text{where} \ h(x) \neq 0\end{align*}

An asymptote is a straight line to which a curve gets closer and closer but never intersects. Asymptotes can be vertical, horizontal, or oblique. This text will focus on vertical asymptotes; other math courses will also show you how to find horizontal and oblique asymptotes.

A vertical asymptote occurs when a function is undefined. A function is undefined when the denominator of a fraction is zero. To find the vertical asymptotes, find where the denominator of the rational function is zero. These are called points of discontinuity of the function.

Rational functions can also have horizontal asymptotes. The equation of a horizontal asymptote is y=c\begin{align*}y=c\end{align*}, where c\begin{align*}c\end{align*} represents the vertical shift of the rational function.

#### Let's find the points of discontinuity and the vertical asymptote for the following function:

y=6x5\begin{align*}y=\frac{6}{x-5}\end{align*}

First, find the value of x\begin{align*}x\end{align*} for which the denominator of the rational function is zero.

0=x5x=5\begin{align*}0=x-5 \rightarrow x=5\end{align*}

The point at which x=5\begin{align*}x=5\end{align*} is a point of discontinuity. Therefore, the asymptote has the equation x=5\begin{align*}x=5\end{align*}.

Look at the graph of the function. There is a clear separation of the branches at the vertical line five units to the right of the origin.

The domain is “all real numbers except five” or symbolically written, x5\begin{align*}x \neq 5\end{align*}.

#### Now, let's identify the vertical and horizontal asymptotes of the following function:

f(x)=3(x4)(x+8)5\begin{align*}f(x)=\frac{3}{(x-4)(x+8)}-5\end{align*}

The vertical asymptotes occur where the denominator is equal to zero.

x4x+8=0x=4=0x=8\begin{align*}x-4 &= 0 \rightarrow x=4\\ x+8 &= 0 \rightarrow x=-8\end{align*}

The vertical asymptotes are x=4\begin{align*}x=4\end{align*} and x=8\begin{align*}x=-8\end{align*}.

The rational function has been shifted down five units: f(x)=3(x4)(x+8)5\begin{align*}f(x)=\frac{3}{(x-4)(x+8)}-5\end{align*}.

Therefore, the horizontal asymptote is y=5\begin{align*}y=-5\end{align*}.

#### Finally, let's solve the following real-world problem:

Electrical circuits are commonplace is everyday life. For instance, they are present in all electrical appliances in your home. The figure below shows an example of a simple electrical circuit. It consists of a battery that provides a voltage (V\begin{align*}V\end{align*}, measured in Volts), a resistor (R\begin{align*}R\end{align*}, measured in ohms, Ω\begin{align*}\Omega\end{align*}) that resists the flow of electricity, and an ammeter that measures the current (I\begin{align*}I\end{align*}, measured in amperes, A\begin{align*}A\end{align*}) in the circuit. Your light bulb, toaster, and hairdryer are all basically simple resistors. In addition, resistors are used in an electrical circuit to control the amount of current flowing through a circuit and to regulate voltage levels. One important reason to do this is to prevent sensitive electrical components from burning out due to too much current or too high a voltage level. Resistors can be arranged in series or in parallel.

For resistors placed in a series, the total resistance is just the sum of the resistances of the individual resistors.

Rtot=R1+R2\begin{align*}R_{tot} =R_1+R_2\end{align*}

For resistors placed in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the resistances of the individual resistors.

1Rc=1R1+1R2\begin{align*}\frac{1}{R_c}=\frac{1}{R_1}+\frac{1}{R_2}\end{align*}

Ohm’s Law gives a relationship between current, voltage, and resistance. It states that:

I=VR\begin{align*}I=\frac{V}{R}\end{align*}

Find the value of x\begin{align*}x\end{align*} marked in the diagram.

Using Ohm’s Law, I=VR\begin{align*}I=\frac{V}{R}\end{align*}, and substituting the appropriate information yields:

2=12R\begin{align*}2= \frac{12}{R}\end{align*}

Using the cross multiplication of a proportion yields:

2R=12R=6 Ω\begin{align*}2R=12 \rightarrow R=6 \ \Omega\end{align*}

### Examples

#### Example 1

Earlier, you were asked about the graph of the function y=85x\begin{align*}y= \frac{8}{5-x}\end{align*}. Susan and Victor both graphed it on a calculator, but Susan thought it looked wrong because it was missing at hole at x=5\begin{align*}x=5\end{align*}. Victor disagreed and said that there is no way the calculator is wrong. Who is right?

When you graph the function y=85x\begin{align*}y= \frac{8}{5-x}\end{align*}, you get the following graph:

As you can see, Susan was correct. There is a gap in the graph at x=5\begin{align*}x=5\end{align*}, so the calculator was wrong.

#### Example 2

Determine the asymptotes of t(x)=2(x2)(x+3)\begin{align*}t(x)=\frac{2}{(x-2)(x+3)}\end{align*}.

Using the Zero Product Property, there are two cases for asymptotes, where each set of parentheses equals zero.

x2x+3=0x=2=0x=3\begin{align*}x-2 &= 0 \rightarrow x=2\\ x+3 &= 0 \rightarrow x=-3\end{align*}

The two asymptotes for this function are x=2\begin{align*}x=2\end{align*} and x=3\begin{align*}x=-3\end{align*}.

Check your solution by graphing the function.

The domain of the rational function above has two points of discontinuity. Therefore, its domain cannot include the numbers 2 or –3. The following is the domain: x2,x3\begin{align*}x \neq 2, x \neq -3\end{align*}.

### Review

1. What is a rational function?
2. Define asymptote. How does an asymptote relate algebraically to a rational equation?
3. Which asymptotes are described in this Concept? What is the general equation for these asymptotes?

Identify the vertical and horizontal asymptotes of each rational function.

1. y=4x+2\begin{align*}y=\frac{4}{x+2}\end{align*}
2. f(x)=52x6+3\begin{align*}f(x)=\frac{5}{2x-6}+3\end{align*}
3. y=10x\begin{align*}y=\frac{10}{x}\end{align*}
4. g(x)=44x2+12\begin{align*}g(x)=\frac{4}{4x^2+1}-2\end{align*}
5. \begin{align*}h(x)=\frac{2}{x^2-9}\end{align*}
6. \begin{align*}y=\frac{1}{x^2+4x+3}+\frac{1}{2}\end{align*}
7. \begin{align*}y=\frac{3}{x^2-4}-8\end{align*}
8. \begin{align*}f(x)=\frac{-3}{x^2-2x-8}\end{align*}

Graph each rational function. Show the vertical asymptote and horizontal asymptote as a dotted line.

1. \begin{align*}y=-\frac{6}{x}\end{align*}
2. \begin{align*}y=\frac{x}{2-x^2}-3\end{align*}
3. \begin{align*}f(x)=\frac{3}{x^2}\end{align*}
4. \begin{align*}g(x)=\frac{1}{x-1}+5\end{align*}
5. \begin{align*}y=\frac{2}{x+2}-6\end{align*}
6. \begin{align*}f(x)=\frac{-1}{x^2+2}\end{align*}
7. \begin{align*}h(x)=\frac{4}{x^2+9}\end{align*}
8. \begin{align*}y=\frac{-2}{x^2+1}\end{align*}
9. \begin{align*}j(x)=\frac{1}{x^2-1}+1\end{align*}
10. \begin{align*}y=\frac{2}{x^2-9}\end{align*}
11. \begin{align*}f(x)=\frac{8}{x^2-16}\end{align*}
12. \begin{align*}g(x)=\frac{3}{x^2-4x+4}\end{align*}
13. \begin{align*}h(x)=\frac{1}{x^2-x-6}-2\end{align*}

Find the quantity labeled \begin{align*}x\end{align*} in the following circuit.

Mixed Review

1. A building 350 feet tall casts a shadow \begin{align*}\frac{1}{2}\end{align*}mile long. How long is the shadow of a person five feet tall?
2. State the Cross Product Property.
3. Find the slope between (1, 1) and (–4, 5).
4. The amount of refund from soda cans in Michigan is directly proportional to the number of returned cans. If you earn a \$12.00 refund for 120 cans, how much do you get per can?
5. You put the letters from VACATION into a hat. If you reach in randomly, what is the probability you will pick the letter \begin{align*}A\end{align*}?
6. Give an example of a sixth-degree binomial.

To see the Review answers, open this PDF file and look for section 12.2.

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### Vocabulary Language: English Spanish

point of discontinuity

A point where the denominator of the rational function is zero. These are used to find the asymptotes of the function.

compression

A stretch or compression is a function transformation that makes a graph narrower or wider, without translating it horizontally or vertically.

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.