Section 4.2 Functions Arising from the Products of Two Lines
Subsection 4.2.1 Overview
You have likely noticed that we are now investigating a new type of function. The graphs have an obvious curve to them; they are not straight lines. As you investigate products of linear functions, think about Batty Functions and what you remember about non-linear functions from your previous work in mathematics courses.Student Page 4.2.2 Investigating Products of Linear Functions
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
(b)
Plot the points
(c)
Connect the points to show the graph of
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2.
(a)
What observations can you make about the graph of
(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
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
(c)
How are the
Student Page 4.2.3 Multiplying Lines Electronically With Desmos
1.
Set up Desmos to play with products of linear functions, as follows:
(a)
Type
(b)
Type
(c)
Type
(d)
Type
2.
Play with the
(a)
How does the graph of the product of two lines change as you change
(b)
Why does the change in
(c)
What happens to the product graph with
(d)
From your previous mathematical experiences, what is the graph of the product called?
3.
Play with the
4.
Edit the first input line to be
(a)
For what values of
(b)
For Task 4.2.3.4.a, is the parabola wider than
(c)
For what values of
(d)
Explain what
5.
Turn off
6.
Continute to play with all three sliders,
(a)
Try each of these three possibilities:
(i)
Both lines have positive slopes.
(ii)
Both lines have negative slopes. (What additional change do you need to make?)
(iii)
One line has a positive slope and the other line has a negative slope.
(b)
How can you tell from the lines what a parabola created from the product of the lines will open upward? Reason from the equations of the lines and the parabola.
(c)
How can you tell from the lines what a parabola created from the product of the lines will open downward? Reason from the equations of the lines and the parabola.
(d)
How can you tell from the lines where the parabola will intersect the
(e)
How can you tell from the lines where the parabola will intersect the
(f)
What is a line of symmetry?
(g)
How can you find the line of symmetry for a function that is the product of two lines? Why does this make sense?
(h)
Each parabola has a highest or lowest point, called the vertex. What information about the vertex can you determine from the lines? Explain.
Activity 9. Polygraph for Parabolas.
Now that we have learned something about how quadratic functions arise in context and their connection to linear functions, let's spend some time describing them. You teacher will provide a Class Code for the Desmos activity, Polygraph: Parabolas in the Quadratic Bundle on Teacher Desmos. You will be randomly paired with another classmate. Polygraph is a variation of the commercial game, Guess Who?. Play 2 rounds of the game or as directed by your teacher.
When time is called, your teacher will provide you another Class Code, this time for the Desmos activity, Polygraph: Parabolas, Part 2. Complete the activity, taking notes on vocabulary that will help you describe and discuss quadratic functions and their graphs more efficiently.
Homework 4.2.4 Homework
1.
Study the graphs in Figure 4.2.4.1. Answer Task 4.2.4.1.a–4.2.4.1.f for the lines provided.
(a)
Complete Table 4.2.4.2. Do not estimate any of the coordinates.
(b)
What is the
for
for
for
(c)
How are the
(d)
From your table, draw an accurate graph of
(e)
The vertex of a parabola is its highest or lowest point. Find the vertex for the function
(f)
Write an equation for each function. Do not simplify the equation for
2.
Table 4.2.4.3 includes several quadratic functions, each provided as a product of two linear factors. Most of the equations are in the form,
Factored Form, |
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Vertex, |
Opens (Up or Down) |
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A | ||||||
B | ||||||
C | ||||||
D | ||||||
E | ||||||
F |
(a)
Find the
(b)
Record the direction the parabola opens, Up or Down.
(c)
The final equation is not yet in factored form though it shows two factors. Write the equation in factored form. What information about the quadratic function is more evident in factored form than in the form provided in the table?
3.
(a)
How can you find the
(b)
How can you tell from the factored form of a quadratic function whether or not the graph of the product will open upward?
(c)
How is the
(d)
How can you find the vertex from the factored form of a quadratic function?
4.
Each quadratic function in Table 4.2.4.4 is given in standard form,
Standard Form, |
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Vertex, |
Opens (Up or Down) |
Factored Form, |
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A | |||||||
B | |||||||
C | |||||||
D | |||||||
E | |||||||
F |
(a)
Graph the quadratic function.
(b)
Complete all but the last column of Table 4.2.4.4. Record points as ordered pairs.
(c)
Suppose one
(d)
Determine a linear equation that shares one of the
(e)
Graph the product of the lines you found in Task 4.2.4.4.d. How does the product compare with the graph of the quadratic function given in standard form? If the graphs are not the same, adjust this equation so the graphs coincide. Compare the equation in factored form with the equation in standard form. What do these equations have in common?
(f)
Two of the equations in the first table could have been factored differently than they were. Study both examples to see the possibilities:
In each example, the first version is considered to be the factored form of the equation. For each equation in this table, determine the factored form and write it in the table. Describe how to find the factored form of a quadratic function.
5.
(a)
Some students say that the vertex of a parabola coincides with the interception of the two lines are multiplied together to find the quadratic function. Is this a true statement? Why or why not?
(b)
Some students remember from previous mathematics classes that the line of symmetry for a quadratic function can be found using the expression,
(c)
Are there any quadratic functions that cannot be factored? If not, why not? If so, give three examples that cannot be factored and explain why they are not factorable.
6.
(a)
How many quadratic functions can share the points
(b)
How many points must you know in order to fit them with a quadratic function? Why do you think so?
(c)
Suppose three points all lie on the same line. Is it reasonable to exactly fit the points with a quadratic function? Why or why not?
7.
Find the factored form of the equation,
8.
Study the parabolas in Figure 4.2.4.5.
(a)
If possible, find an equation for each graph. Explain how you know each equation is correct.
(b)
If you cannot find an equation for a graph, explain why it is not possible with your current knowledge of quadratic functions.