2.4 2.52.32.1 2.2

Up and Down the StaircaseExploring Slope 2.2

Linear functions are often used as models for patterns in data plots. In Moving Straight Ahead, you learned several facts about equations representing linear functions.

• Any linear function can be expressed by an equation in the form y = mx + b.

• The value of the coefficient m tells the rate at which the values of y increase (or decrease) as the values of x increase by 1. Since m tells you the change in y for every one-unit change in x, it can also be called the unit rate. A unit rate is a rate in which the second number is 1, or 1 of a quantity.

• The value of m also tells the steepness and direction (upward or downward) of the graph of the function.

• The value of b tells the point at which the graph of the function crosses the y-axis. That point has coordinates (0, b) and is called the y-intercept.

In any problem that calls for a linear model, the goal is to find the values of m and b for an equation with a graph that fits the data pattern well. To measure the steepness of a linear equation graph, it helps to imagine a staircase that lies underneath the line.

horizontal change

vertical change

Thinking With Mathematical Models34

2.4 2.52.32.1 2.2

The steepness of a staircase is commonly measured by comparing two numbers, the rise and the run. The rise is the vertical change from one step to the next, and the run is the horizontal change from one step to the next.

y

x

20 in.

8 in.

12 in.

The steepness of the line is the ratio of rise to run. This ratio is the slope of the line.

slope =vertical change

horizontal change=

riserun

In the diagram of the staircase, the slope of the line is 23. The y-intercept is (0, 20). So the equation for the linear function is y = 2

3 x + 20.

Investigation 2 Linear Models and Equations 35

2.4 2.52.32.1 2.2

Use the data given in each question to find the equation of the linear function relating y and x.

A For the functions with the graphs below, find the slope and y-intercept. Then write the equations for the lines in the form y = mx + b.

1.

O−4 −2 2 4

−4

−2

2

4y

x

2.

O−4 −2 2 4

−4

−2

2

4y

x

3.

O−4 −2 2 4

−4

−2

2

4y

x

4. y

O−4 −2 2 4

−4

−2

2

4y

x

2.2Problem

continued on the next page >

Thinking With Mathematical Models36

2.4 2.52.32.1 2.2 2.4 2.52.32.1 2.2

B 1. Find equations for the linear functions that give these tables. Write them in the form y = mx + b.

a. 20 1-1-2

7531-1

x

y

b. 102 6-2-6

420-2-4

x

y

2. For each table, find the unit rate of change of y compared to x.

3. Does the line represented by this table have a slope that is greater than or less than the equations you found in part 1(a) and part 1(b)?

x

y

31 20-1

-8-5-214

C The points (4, 2) and (-1, 7) lie on a line.

1. What is the slope of the line?

2. Find two more points that lie on this line. Describe your method.

3. Yvonne and Jackie observed that any two points on a line can be used to find the slope. Are they correct? Explain why or why not.

D Kevin said that the line with equation y = 2x passes through the points (0, 0) and (1, 2). He also said the line with equation y = -3x passes through the points (0, 0) and (1, -3). In general, lines with equations of the form y = mx always pass through the points (0, 0) and (1, m). Is he correct? Explain.

E What is the slope of a horizontal line? Of a vertical line?

2.2Problem continued

Homework starts on page 45.

Investigation 2 Linear Models and Equations 37