Lesson 2 You-Mix Cubes Solidify Understanding

Jump Start

1.

Identify the key features of $g (x) = x^{2}$ , including the domain, range, $x$ and $y$ intercepts, and intervals of increase and decrease.

2.

Graph each of the following functions using transformations of $g (x) = x^{2}$ .

a.

$y = x^{2} + 3$

b.

$y = {(x + 3)}^{2}$

c.

$y = - {(x - 1)}^{2} + 2$

Learning Focus

Compare quadratic and cubic functions.

Graph cubic functions.

What are the similarities and differences between quadratic functions and cubic functions?

Open Up the Math: Launch, Explore, Discuss

Previously, we learned that the function generated by the sum of the terms in a quadratic sequence is called a cubic function. Linear functions, quadratic functions, and cubic functions are polynomials, which also include functions of higher powers. In this lesson, we will explore more about cubic functions to learn some of the similarities and differences between cubic functions and quadratic functions.

To begin, let’s take a look at the most basic cubic function, $f (x) = x^{3}$ . It is technically a degree $3$ polynomial because the highest exponent is $3$ , but it’s called a cubic function because these functions are often used to model volume. Similarly, quadratic functions are degree $2$ polynomials but are called quadratic after the Latin word for square. Scott’s March Motivation showed that linear functions have a constant rate of change, quadratic functions have a linear rate of change, and cubic functions have a quadratic rate of change.

1.

Use a table to verify that $f (x) = x^{3}$ has a quadratic rate of change.

2.

Graph $f (x) = x^{3}$ .

3.

Describe the features of $f (x) = x^{3}$ : including intercepts, intervals of increase or decrease, domain, range, etc.

4.

Using your knowledge of transformations, graph each of the following without using technology.

a.

$f (x) = x^{3} - 3$

b.

$f (x) = {(x + 3)}^{3}$

c.

$f (x) = 2 x^{3}$

d.

$f (x) = - {(x - 1)}^{3} + 2$

5.

Use technology to check your graphs above. What transformations did you get right? What areas do you need to improve on so that your cubic graphs are perfect?

6.

Since quadratic functions and cubic functions are both in the polynomial family of functions, we would expect them to share some common characteristics. List all the similarities between $f (x) = x^{3}$ and $g (x) = x^{2}$ .

7.

As you can see from the graph of $f (x) = x^{3}$ , there are also some real differences in cubic functions and quadratic functions. Each of the following statements describes one of those differences. Explain why each statement is true by completing the sentence.

a.

The range of $f (x) = x^{3}$ is $(- \infty, \infty)$ , but the range of $g (x) = x^{2}$ is $(0, \infty)$ because:

b.

For $x > 1$ , $f (x) > g (x)$ because:

c.

For $0 < x < 1, g (x) > f (x)$ because:

Now that you have some understanding of the graphs of cubic functions, let’s relate cubic functions to their inverses, cube root functions.

8.

The inverse of $f (x) = x^{3}$ is $g (x) = \sqrt[3]{x}$ .

a.

Use the graph of $f (x) = x^{3}$ to find the graph of $g (x) = \sqrt[3]{x}$ .

b.

What do you observe about the anchor points of $f (x) = x^{3}$ and $g (x) = \sqrt[3]{x}$ ?

9.

Examine your work on problem 4d, where $f (x) = - {(x - 1)}^{3} + 2$

a.

Graph $f^{- 1} (x)$

b.

Write the equation of $f^{- 1} (x) .$

c.

Use technology to check your work. Make the adjustments needed to correct your work, if any.

d.

How does $f^{- 1} (x)$ compare with $g (x) = \sqrt[3]{x}$ ?

10.

Given $f (x) = 2 \sqrt[3]{(x + 3)}$

a.

Graph $f (x)$

b.

Find the equation of $f^{- 1} (x)$ .

Ready for More?

What similarities do you expect to exist between $f (x) = x^{3}$ and $k (x) = x^{5}$ ?

Takeaways

*Similarities between $f (x) = x^{3}$ and $g (x) = x^{2}$ .	Differences in $f (x) = x^{3}$ and $g (x) = x^{2}$ .

*The similarities between $f (x) = x^{3}$ and $g (x) = x^{2}$ are true of all polynomial parent functions, $h (x) = x^{4}$ , $k (x) = x^{5}$ , etc.

Vocabulary

cubic function
degree of a polynomial
polynomial function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined the features of the cubic parent function, $f (x) = x^{3}$ . We identified anchor points and learned to use transformations to graph functions in the form $y = a {(x - h)}^{3} + k$ . We compared $f (x) = x^{3}$ and $g (x) = x^{2}$ to identify similarities and differences in domain, range, intercepts, and intervals of increase and decrease.

Retrieval

1.

a.

Use the graph to write the equation of the quadratic function in factored form.

b.

How can you use the equation to find the $y$ -intercept?

2.

Write an equivalent expression: $x^{\frac{5}{18}} \cdot x^{\frac{4}{9}} \cdot x^{\frac{1}{3}}$ .