Graphing Linear Relationships

Learning Objectives

Identify a linear relationship from a table of values.

Understand the meaning of slope as the rate of change.

Understand the meaning of the y-intercept as the starting value.

Graph a linear relationship given an equation in the form y = mx + b.

Lines on the Plane

In the last lesson, we learned how to plot individual points on the coordinate plane. Now, let's look at what happens when points form a pattern, specifically a straight line. A relationship that creates a straight line on a graph is called a linear relationship.

Constant Rate of Change

The defining feature of a linear relationship is a constant rate of change. This means that for every step you take in the x-direction, you take a constant-sized step in the y-direction.

Consider this table of values for x and y:

| x | y |

|:---:|:---:|

| 0 | 1 |

| 1 | 3 |

| 2 | 5 |

| 3 | 7 |

Notice that every time x increases by 1, y increases by 2. This constant rate of change (change in y / change in x = 2/1 = 2) tells us the relationship is linear.

Slope-Intercept Form: y = mx + b

The most common way to write the equation for a line is the slope-intercept form.

y = mx + b

This form is powerful because it tells us everything we need to know to graph the line.

b is the y-intercept. This is the point where the line crosses the vertical y-axis. It's the value of y when x is 0. Think of it as the starting point.

m is the slope. The slope is a number that measures the 'steepness' of the line. It is the rate of change. The slope is often written as a fraction: rise / run.

Rise: How many units the line goes up (positive) or down (negative).

Run: How many units the line goes to the right.

How to Graph a Line from the Equation

Let's graph the equation y = (2/3)x + 1.

1.Begin with 'b'. The y-intercept (b) is 1. This means the line crosses the y-axis at the point (0, 1). Find this point on your graph and plot it. This is your starting point.

2.Move with 'm'. The slope (m) is 2/3. This means the rise is 2 and the run is 3.

3.Plot the next point. Starting from your y-intercept (0, 1), 'rise' 2 units (go up 2) and then 'run' 3 units (go right 3). You will land on the point (3, 3). Plot this point.

4.Repeat if needed. You can do this again from your new point (3, 3). Rise 2 and run 3. You will land on the point (6, 5).

5.Draw the line. Use a ruler to connect the points you plotted. Draw arrows on the ends to show that the line continues forever. You have now graphed the equation!

Key Terms

**Linear Relationship: A relationship between two variables that, when plotted on a graph, produces a straight line. It has a constant rate of change.
**Slope (m): A measure of the steepness of a line
**y-intercept (b): The point where a line crosses the y-axis
**Rate of Change: A ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.
**Slope-Intercept Form: A common form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept.