Lesson 1 Winner, Winner Develop Understanding

Learning Focus

Understand the behavior of $f (x) = \frac{1}{x}$ for very large $x$ values and for $x$ values near $0$ .

Graph and describe the features of $f (x) = \frac{1}{x}$ using appropriate notation.

How can winning the lottery help us to think about $f (x) = \frac{1}{x}$ ?

What features make $f (x) = \frac{1}{x}$ mathematically interesting?

Open Up the Math: Launch, Explore, Discuss

One of the most interesting functions in mathematics is $f (x) = \frac{1}{x}$ because it brings up some mathematical mind benders. In this task, we will use a story context and representations like tables and graphs to understand this important function.

Let’s begin by thinking about the interval $[1, \infty)$ .

1.

Imagine that you won the lottery and were given one big pot of money. Of course, you would want to share the money with friends and family. If you split the money evenly between yourself and one friend, what would be each person’s share of the prize money?

2.

If three people shared the prize money, what would be each person’s share?

3.

Model the situation with a table, equation, and graph.

4.

Just in case you didn’t think about the really big numbers in your model, how much of the prize would each person get if $1, 000$ people get a share? If $100, 000$ people get a share? If $100, 000, 000$ people get a share?

5.

Use mathematical notation to describe the behavior of this function as $x \to \infty$ .

Next, let’s look at the interval $(0, 1]$ and consider a new way to think about splitting the prize money.

6.

Imagine that you want each person’s share to be $\frac{1}{2}$ of the prize. How many people could share the prize?

7.

If you want each person’s share to be $\frac{1}{3}$ of the prize, how many people could share the prize?

8.

Model this situation with a table, equation, and graph.

9.

What do you notice when you compare the two models that you have written?

Now, let’s put it all together to graph the entire function, $f (x) = \frac{1}{x}$ .

10.

Create a table for $f (x) = \frac{1}{x}$ that includes negative input values.

11.

How do the values of $f (x)$ in the interval from $(- \infty, 0)$ compare to the values of $f (x)$ from $(0, \infty)$ ? Use this comparison to predict the graph of $f (x) = \frac{1}{x}$ .

12.

Graph $f (x) = \frac{1}{x}$ .

13.

Describe the features of $f (x) = \frac{1}{x}$ , including domain, range, intervals of increase or decrease, $x$ - and $y$ - intercepts, end behavior, and any maximum(s) or minimum(s).

Ready for More?

How would you explain to a friend why $\frac{1}{0}$ is not $0$ ? What reasons can you give for why $\frac{1}{0}$ is undefined?

Takeaways

Features of $f (x) = \frac{1}{x}$

Interval $(1, \infty)$
Interval $(0, 1]$
Characteristics
Symmetry
End behavior
Asymptotes
Domain
Range
Intervals of decrease
Intercepts, maximum, minimum

Vocabulary

asymptote
horizontal asymptote
vertical asymptote
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about the function $f (x) = \frac{1}{x}$ , a rational function. We learned about the features of the function and its behavior near the horizontal and vertical asymptotes.

Retrieval

1.

Describe the transformation of this function from $f (x) = x^{2}$ . Then write the equation in vertex form.

2.

Write the equation of this exponential function. Then write the equation of the horizontal asymptote.