Graphs of Rational Functions

Graphs of Rational Functions

• Let’s explore graphs and equations of rational functions.

• Let’s learn about horizontal asymptotes.

I can identify a vertical asymptote from a graph or an equation of a rational function.

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​Learning Targets

Rational Function

A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.

vertical asymptote

The line x = a is a vertical asymptote for a function f if f is undefined at x = a and its outputs get larger and larger in the negative or positive direction when x gets closer and closer to a on each side of the line. This means the graph goes off in the vertical direction on either side of the line.

Vertical asymptotes come from the factors of the denominator that are not in common with a factor of the numerator. The vertical asymptotes occur where those factors equal zero.

Identify Vertical Asymptotes of a Rational Function

1. Factor the numerator and denominator.

2. Simplify by canceling common factors in the numerator and the denominator.

3. Set the simplified denominator equal to zero and solve for x.

horizontal asymptote

The line The line y = c  is a horizontal asymptote of a function if the outputs of the function get closer and closer to c as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the x-axis.

Horizontal Asymptotes of Rational Functions

The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.

If N is the degree of the numerator and D is the degree of the denominator, and…

• N < D, then the horizontal asymptote is y = 0.

• N = D, then the horizontal asymptote is y = ratio of leading coefficients.

• N > D, then there is no horizontal asymptote.

Open Ended

Identify the horizontal asymptote. $$g\left(x\right)=\frac{x^2+5x-4}{2x^2-16}$$

Open Ended

Kiran’s aunt plans to bike 10 miles. How long will it take if she bikes at an average rate of 8 miles per hour?

Open Ended

Kiran’s aunt plans to bike 10 miles. How long will it take if she bikes at an average rate of r miles per hour?

Open Ended

Kiran’s aunt plans to bike 10 miles. Kiran wants to join his aunt, but he only has 45 minutes to exercise. What will their average rate need to be for him to finish on time?

Open Ended

Kiran plans to bike 10 miles. Write an equation that gives his time t, in hours, as a function of his rate r, in miles per hour. What would the graph look like?

Open Ended

Kiran plans to bike 10 miles. What is the meaning of t(8)? Does this value make sense? Explain your reasoning.

Open Ended

Kiran plans to bike 10 miles. What is the meaning of t(0)? Does this value make sense? Explain your reasoning.

Open Ended

As r gets closer and closer to 0, what does the behavior of the function tell you about the situation?

Open Ended

f and g are both rational functions defined by $$f\left(x\right)=\frac{6}{x}$$ and $$g\left(x\right)=\frac{6}{x-1}$$ .

Here are their graphs. 

What do you notice? What do you wonder?

Open Ended

Let c be the function that gives the average cost per book c(x), in dollars, when using an online store to print x copies of a self-published paperback book. Here is a graph of   $$c\left(x\right)=\frac{120+4x}{x}$$ .

What is the approximate cost per book when 50 books are printed? 100 books?

Open Ended

Let c be the function that gives the average cost per book c(x), in dollars, when using an online store to print x copies of a self-published paperback book. Here is a graph of   $$c\left(x\right)=\frac{120+4x}{x}$$ .

The author plans to charge $8 per book. About how many should be printed to make a profit?

Open Ended

Let c be the function that gives the average cost per book c(x), in dollars, when using an online store to print x copies of a self-published paperback book. Here is a graph of   $$c\left(x\right)=\frac{120+4x}{x}$$ .

What does the end behavior of the function say about the context?

Multiple Choice

Match the graph with it's corresponding functions.

• $$c\left(x\right)=\frac{1+4x}{x},\ h\left(x\right)=\frac{1}{x}+4$$

• $$e\left(x\right)=\frac{1-4x}{x},\ b\left(x\right)=\frac{1}{x}-4$$

• $$d\left(x\right)=\frac{x+4}{x},\ g\left(x\right)=1+\frac{4}{x}$$

• $$f\left(x\right)=\frac{4-x}{x},\ a\left(x\right)=\frac{4}{x}-1$$

Multiple Choice

Match the graph with it's corresponding functions.

• $$c\left(x\right)=\frac{1+4x}{x},\ h\left(x\right)=\frac{1}{x}+4$$

• $$e\left(x\right)=\frac{1-4x}{x},\ b\left(x\right)=\frac{1}{x}-4$$

• $$d\left(x\right)=\frac{x+4}{x},\ g\left(x\right)=1+\frac{4}{x}$$

• $$f\left(x\right)=\frac{4-x}{x},\ a\left(x\right)=\frac{4}{x}-1$$

Multiple Choice

Match the graph with it's corresponding functions.

• $$c\left(x\right)=\frac{1+4x}{x},\ h\left(x\right)=\frac{1}{x}+4$$

• $$e\left(x\right)=\frac{1-4x}{x},\ b\left(x\right)=\frac{1}{x}-4$$

• $$d\left(x\right)=\frac{x+4}{x},\ g\left(x\right)=1+\frac{4}{x}$$

• $$f\left(x\right)=\frac{4-x}{x},\ a\left(x\right)=\frac{4}{x}-1$$

Multiple Choice

Match the graph with it's corresponding functions.

• $$c\left(x\right)=\frac{1+4x}{x},\ h\left(x\right)=\frac{1}{x}+4$$

• $$e\left(x\right)=\frac{1-4x}{x},\ b\left(x\right)=\frac{1}{x}-4$$

• $$d\left(x\right)=\frac{x+4}{x},\ g\left(x\right)=1+\frac{4}{x}$$

• $$f\left(x\right)=\frac{4-x}{x},\ a\left(x\right)=\frac{4}{x}-1$$

Open Ended

The average cost for printing x copies of a self-published paperback book with Company A is $$c\left(x\right)=\frac{120+4x}{x}$$ , The average cost for printing  copies of a paperback book with Company B is $$d\left(x\right)=\frac{25+10x}{2x}$$

Which company would you recommend to an author who wants to print 100 books? Explain your reasoning.

Open Ended

The average cost for printing x copies of a self-published paperback book with Company A is $$c\left(x\right)=\frac{120+4x}{x}$$ , The average cost for printing  copies of a paperback book with Company B is $$d\left(x\right)=\frac{25+10x}{2x}$$

Which company would you recommend to an author who thinks their book will be a best seller and needs to print thousands of books? How could you rewrite the equations to make the choice clearer?

I can identify a vertical asymptote from a graph or an equation of a rational function.

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​Learning Targets

Rational Expressions, Vertical Asymptotes, and Holes

I can Find Asymptotes and Holes of Rational Equations

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​Learning Targets

A hole is a point on the graph where the value of the function is not defined. If the numerator and denominator of a rational function have a common factor, they will cancel when simplifying. The cancelled value creates a hole in the graph. To determine the x -coordinate of a hole, set the cancelled factor equal to zero and solve. To determine the y -coordinate, plug the x -value into the simplified function and solve.

​Holes in Rational Functions

​https://youtu.be/GW6O7jyhpoQ