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Student Page 2.6.4 Finding Equations from Graphs and Contexts, Interpreting Slope and y-Intercept
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1.

Compare the graphs in Figure 2.6.4.1. Which ones are related to each other? How are they related? (Note: The green line is Graph A, blue is Graph B, black is Graph C, and red is Graph D)

Figure 2.6.4.1. Four Linear Plots

2.

(a)

Complete a table for each graph. Use Table 2.6.4.2, Table 2.6.4.3, Table 2.6.4.4, and Table 2.6.4.5.

Table 2.6.4.2. Graph A (Green)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
Table 2.6.4.3. Graph B (Blue)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
Table 2.6.4.4. Graph C (Black)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
Table 2.6.4.5. Graph D (Red)
\(x\) \(y\)
0    
1    
2    
3    
4    
5    
6    
(b)

Determine a recursive pattern for each table.

(c)

Use the recursive rule to find an explicit equation for each graph. Write the recursive pattern and the equation under each table.

3.

(b)

How can you find the slope of a graph from a table?

(c)

How can you find the slope of a graph from an equation?

(d)

How can you find the slope directly from a graph? Show how the slope arises from the graph in each case.

4.

(a)

What is the \(y\)-intercept?

(b)

How can you find the \(y\)-intercept from a table?

(c)

How can you find the \(y\)-intercept from an equation?

(d)

How can you find the \(y\)-intercept from a graph?

5.

Consider the contexts that follow.

Context A

Raffle tickets are $0.50 each. How much will you pay for \(T\) tickets?

Context B

You have $6 to buy ride tickets at a fair. Each ticket costs $1. How much money do you have after you purchase \(T\) tickets?

Context C

It costs $2 to get into a dance. At the dance you can buy glasses of punch for $0.50 each. How much will you spend for the dance and punch after buying \(G\) glasses of punch?

Context D

The Rowing Team is selling cookies for $1 each. A friend is selling the cookies and allows you to pay later for cookies you buy beyond the $3 you have in your pocket. How much will you pay your friend if you buy \(C\) cookies? (Negative dollars are dollars you will pay your friend after the cookie sale.)

(b)

Graph the data in each table.

7.

Match the contexts in Student Page Exercise 2.6.4.6 to the graphs in Student Page Exercise 2.6.4.1. How do you know you are right?

\begin{align*} \amp \text{Graph A} \amp \qquad \amp \text{Context A}\\ \amp \text{Graph B} \amp \qquad \amp \text{Context B}\\ \amp \text{Graph C} \amp \qquad \amp \text{Context C}\\ \amp \text{Graph D} \amp \qquad \amp \text{Context D} \end{align*}

8.

Some students say that you can find the slope from a table by dividing the value of \(y\) by the value of \(x\) for a single data point.

(a)

Does this method ever work? If so, for which of the examples above will this method work? Why does the method work?

(b)

Try this method for finding slope on Table 2.6.4.3. Does the method work in this case? Why or why not?

(c)

For which of the examples above will this method for finding slope fail? Why does it fail?