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?