### Student Page 4.2.2 Investigating Products of Linear Functions

Solve the problems that follow to investigate the appearance of products of linear functions. Think about these questions as you work: To what family of functions do products of linear functions belong? What shape is the graph of this function family? (Note: For a review of function notation, please see A Word about Function Notation)

#### 1.

Graphs of lines are provided in Figure 4.2.2.1, Figure 4.2.2.2, and Figure 4.2.2.3. Complete the following for each pair of graphs:

##### (a)

For each value of \(x\) in Table 4.2.2.4, Table 4.2.2.5 or Table 4.2.2.6, determine the values of \(f(x)\text{,}\) \(g(x)\text{,}\) and the product \(h(x) = f(x) \cdot g(x)\text{.}\)

##### (b)

Plot the points \((x, h(x))\) on the graph in Figure 4.2.2.1, Figure 4.2.2.2, or Figure 4.2.2.3.

##### (c)

Connect the points to show the graph of \(y = h(x)\text{.}\)

\(x\) | \(f(x)\) | \(g(x)\) |
\(h(x) =\) \(f(x) \cdot g(x)\) |
---|---|---|---|

\(-4\) | |||

\(-3\) | |||

\(-2\) | |||

\(-1\) | |||

\(0\) | |||

\(1\) | |||

\(2\) | |||

\(3\) |

\(x\) | \(f(x)\) | \(g(x)\) |
\(h(x) =\) \(f(x) \cdot g(x)\) |
---|---|---|---|

\(-4\) | |||

\(-3\) | |||

\(-2\) | |||

\(-1\) | |||

\(0\) | |||

\(1\) | |||

\(2\) | |||

\(3\) |

\(x\) | \(f(x)\) | \(g(x)\) |
\(h(x) =\) \(f(x) \cdot g(x)\) |
---|---|---|---|

\(-4\) | |||

\(-3\) | |||

\(-2\) | |||

\(-1\) | |||

\(0\) | |||

\(1\) | |||

\(2\) | |||

\(3\) |

#### 2.

##### (a)

What observations can you make about the graph of \(y = h(x)\) based on the graphs of \(y = f(x)\) and \(y = g(x)\text{?}\) Explain why your observations make sense.

##### (b)

What is the shape of the product graph? Why is the product graph shaped this way?

##### (c)

What type of function arises from multiplying two linear functions together?

#### 3.

##### (a)

In order for a product of two numbers to be zero, what do you know about one or both of the numbers you are multiplying together?

##### (b)

A function arising from the product of two lines has an \(x\)-intercept at \(x = 3\text{.}\) The equation is written as a product of two linear functions. What do you know about one of the linear functions? How does this information relate to Task 4.2.2.3.a?

#### 4.

Through investigating the product of two lines, we are able to find an equation of a function from the equations of the lines that are multiplied together to form it.

##### (a)

Find an equation for each of the products of the pairs of lines in Student Page Exercise 4.2.2.1.

##### (b)

How are the \(x\)-intercepts of each line related to the \(x\)-intercepts of the product graph?

##### (c)

How are the \(x\)-intercepts of each function related to its equation?