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Section 2.8 Linear Functions: Big Ideas

Subsection 2.8.1 Overview

Let's take some time to revisit and synthesize what you've been exploring about linear functions from this chapter.

Student Page 2.8.2 Big Ideas ‚ÄĒ Slope and \(y\)-Intercept

Use Desmos on your laptop, tablet, or smart phone. Be ready to share your findings and to demonstrate using the classroom computer.

By now you have discovered that all linear equations can be represented generally by the equation, \(y = mx + b\text{,}\) where \(x\) and \(y\) are variables and \(m\) and \(b\) represent known numbers (\(m\) and \(b\) are called parameters).

1.

Type this general form of a linear equation into Desmos. Create sliders for \(m\) and \(b\text{.}\) For Task 2.8.2.1.a and Task 2.8.2.1.c, use one or two rounds of Roundtable (see Roundtable) to share everything you have learned. Explain each item in your list as you create it.

(a)

Let \(y = mx\) and play with \(m\text{.}\) List everything you can about the effect of \(m\) on the graph of \(y = mx\text{.}\)

(b)

Revisit your list. If you have not already answered the following, do so now. Explain! What happens to the graph of \(y = mx\) when:

(i)

\(m \gt 0\text{?}\)

(ii)

\(m \lt 0\text{?}\)

(iii)

\(\left| m \right|\) is large?

(iv)

\(\left| m \right|\) is close to zero?

(c)

Let \(m = 1\) and play with \(b\text{.}\) List everything you can about the effect of \(b\) on the graph of \(y = mx + b\text{.}\)

(d)

Revisit your list. If you have not already answered the following, do so now. What happens to the graph of \(y = mx + b\) when:

(i)

\(b \gt 0\text{?}\)

(ii)

\(b \lt 0\text{?}\)

(iii)

\(\left| b \right|\) is large?

(iv)

\(\left| b \right|\) is close to zero?

(v)

How is the graph of \(y = mx + b\) moving as you change the value of \(b\text{?}\)

(vi)

Why is the graph moving as you say it does?

2.

(a)

Place a page protector over the screen on your electronic tool. Trace the \(x\)- and \(y\)-axes on the page protector. By hand, graph the following equations. Explain how you drew each graph so that you know it is accurate.

(i)

\(y = 2x + 4\)

(ii)

\(y = 0.5x - 3\)

(iii)

\(y = -x - 1\)

(b)

Once all group members are satisfied with the three hand-drawn graphs from Task 2.8.2.2.a graph them electronically. Was each hand-drawn graph accurate? If not, what needs to be fixed? What were you thinking when you drew each graph? What do you now know? Turn off the electronic graph and redraw the graph using what you now know. Recheck.

3.

Study the graphs. Find an equation for each graph. Do not estimate; use grid points to find each equation accurately. In each case, explain how you know your choices of slope and \(y\)-intercept are correct.

(a)
(b)
(c)

4.

Study the tables. For each table, find an equation to fit the data. In each case, explain how you know your choices of slope and \(y\)-intercept are correct.

(a)
\(x\) \(y\)
4 \(-5\)
8 \(-7\)
12 \(-9\)
16 \(-11\)
20 \(-13\)
(b)
\(x\) \(y\)
8 3
10 11
12 19
14 27
16 35
(c)
\(x\) \(y\)
1922 450
1928 435
1934 420
1940 405
1946 390

5.

Study each context. For each context, find an equation. In each case, explain how you know your choices of slope and \(y\)-intercept are correct.

(a)

Fia earns 3 points every time she finds a star in her Dora the Explorer game. She earns a one-time bonus of 50 points in each level for finding a banana for her monkey friend, Boots. Based on the number of stars she finds, how many points will Fia earn in a level if she always finds the banana?

(b)

Valen earns $5 each week for doing his chores. Each time his mom has to remind him, there is a $0.50 deduction from his allowance. How much will Valen earn in a week based on the number of times he needs to be reminded to do his chores?

(c)

Jackson has $18. Each month, he pays $4 to stream music on his iPod. How much does he have or owe based on the number of months he streams music?

6.

In each case below, explain how you find the slope and the \(y\)-intercept in each representation.

(a) \(y\)-intercept, \(b\).
(i)

Table:

(ii)

Graph:

(iii)

Context:

(b) Slope, \(m\).
(i)

Table:

(ii)

Graph:

(iii)

Context:

Homework 2.8.3 Homework

1.

Study the graph in Figure 2.8.3.1 Answer the following for this line.

Figure 2.8.3.1. Reference for Exercise 2.8.3.1
(a)

What is an intercept? How can you find an intercept on a graph?

(b)

Which point is the \(y\)-intercept? Write the point as an ordered pair, \((x, y)\text{.}\) How do you know this point is the \(y\)-intercept?

(c)

Which point is the \(x\)-intercept? Write the point as an ordered pair, \((x, y)\text{.}\) How do you know this point is the \(x\)-intercept?

(d)

Find five points grid points on the graph. Do not estimate. Write them in the table below.

\(x\) 0        
\(y\)   0      
(e)

Find the slope of the line through these points. Show or explain your work.

(f)

Write an equation for the line.

2.

A line has slope, \(m = 0.75\text{.}\) Its \(y\)-intercept is the point \((0, 2)\text{.}\)

(b)

Write an equation for the line.

3.

The slope of a line is \(-2\text{.}\) One point on the line is \((-3, 4)\text{.}\)

(b)

Find the \(y\)-intercept and the equation of the line.

(c)

How do you know your equation is correct?

4.

Find an equation to fit each data set. Show and explain your work. Fill in the last 2 rows of each table to fit the equation you find.

(a)
\(x\) \(y\)
\(-3\) 5
\(-2\) 11
\(-1\) 17
0 23
1 29
2  
17  
(b)
\(x\) \(y\)
4 17
12 6
20 \(-5\)
28 \(-16\)
36 \(-27\)
44  
100  
(c)
\(x\) \(y\)
120 15
125 18
130 21
135 24
140 27
145  
200  
(d)
\(x\) \(y\)
30 6
32 6.5
34 7
36 7.5
38 8
40  
150  

5.

Write stories for two of the tables in Exercise 2.8.3.4. Explain how each story fits the corresponding table.

Student Page 2.8.4 Distinguishing Linear from Other Functions

1.

What do you know about linear equations? Give brief answers now and think about these questions as you work on the problems below.

(a)

What patterns would you see in a table of data that can be modeled by a linear equation?

(b)

How can you tell a graph can be modeled by a linear equation?

(c)

What does an equation of a linear equation look like?

(d)

How can you tell a story can be modeled by a linear equation?

2.

Study the data sets. Label each data set as linear or non-linear. Explain your choice.

(a)
\(x\) \(y\)
\(-5\) \(-14\)
\(-4\) \(-12\)
\(-3\) \(-10\)
\(-2\) \(-8\)
\(-1\) \(-6\)
0 \(-4\)
1 \(-2\)
2 0
3 2
4 4
5 6

Linear / Non-linear? Explain:

(b)
\(x\) \(y\)
\(-5\) 0.03125
\(-4\) 0.0625
\(-3\) 0.125
\(-2\) 0.25
\(-1\) 0.5
0 1
1 2
2 4
3 8
4 16
5 32

Linear / Non-linear? Explain:

(c)
\(x\) \(y\)
\(-5\) 25
\(-4\) 16
\(-3\) 9
\(-2\) 4
\(-1\) 1
0 0
1 1
2 4
3 9
4 16
5 25

Linear / Non-linear? Explain:

3.

Categorize each graph as linear or non-linear. Explain your choice.

(a)
(for accessibility)

Linear / Non-linear? Explain:

(b)
(for accessibility)

Linear / Non-linear? Explain:

(c)
(for accessibility)

Linear / Non-linear? Explain:

4.

Gather data for each story. Label each data set as linear or non-linear. Explain your choice.

(a)

Geppetto carved Pinocchio a 2-inch long nose. Each time Pinocchio tells a lie, his nose grows 3 inches. How long will his nose be after 1 lie? 2 lies? 3 lies? \(L\) lies?

Lie Number Nose Length in inches
0 2
1  
2  
3  
4  
5  
\(L\)  

Linear / Non-linear? Explain:

(b)

Geppetto carved Pinocchio a 2-inch long nose. Each time Pinocchio tells a lie, his nose doubles in length. How long will his nose be after 1 lie? 2 lies? 3 lies? \(L\) lies?

Lie Number Nose Length in inches
0 2
1  
2  
3  
4  
5  
\(L\)  

Linear / Non-linear? Explain:

(c)

Geppetto carved Pinocchio a 2-inch long nose. Each time Pinocchio tells a lie, his nose grows from its previous length by the same number of inches as the number of lies he has told. For example, after 1 lie, his nose will be 2 + 1 = 3 inches long. After 2 lies, his nose will be 3 + 2 = 5 inches long. How long will his nose be after 3 lies? 4 lies? 5 lies? \(L\) lies?

Lie Number Nose Length in inches
0 2
1 3
2 5
3  
4  
5  
\(L\)  

Linear / Non-linear? Explain:

5.

Categorize each equation as linear or non-linear. Explain your choice.

(a)

\(y = 3x + 4\)

Linear / Non-linear? Explain:

(b)

\(y = 2^x + 3\)

Linear / Non-linear? Explain:

(c)

\(y = x^2 + 1\)

Linear / Non-linear? Explain:

6.

Revisit Exercise 2.8.3.1. What can you add to your previous answers? Use more space as needed.

(a)

What patterns would you see in a table that can be modeled by a linear equation?

(b)

How can you tell a graph can be modeled by a linear equation?

(c)

What does an equation of a linear equation look like?

(d)

How can you tell a story can be modeled by a linear equation?