Lesson 2 You-Mix Cubes Solidify Understanding

Jump Start

1.

Identify the key features of , including the domain, range, and intercepts, and intervals of increase and decrease.

2.

Graph each of the following functions using transformations of .

a.

a blank 17 by 17 grid

b.

a blank 17 by 17 grid

c.

a blank 17 by 17 grid

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, . It is technically a degree polynomial because the highest exponent is , but it’s called a cubic function because these functions are often used to model volume. Similarly, quadratic functions are degree 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 has a quadratic rate of change.

2.

Graph .

a blank 17 by 17 grid

3.

Describe the features of : 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.

a blank 17 by 17 grid

b.

a blank 17 by 17 grid

c.

a blank 17 by 17 grid

d.

a blank 17 by 17 grid

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 and .

7.

As you can see from the graph of , 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 is , but the range of is because:

b.

For , because:

c.

For because:

Ready for More?

What similarities do you expect to exist between and ?

Takeaways

*Similarities between and .

Differences in and .

*The similarities between and are true of all polynomial parent functions, , , etc.

Vocabulary

Lesson Summary

In this lesson, we examined the features of the cubic parent function, . We identified anchor points and learned to use transformations to graph functions in the form . We compared and 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.

A parabola opening up with the points (1,0),(2,-1), and (3,0) graphed on a coordinate plane. x–2–2–2222444666y–2–2–2222444666000

b.

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

2.

Write an equivalent expression: .