Graphing Linear Equations Section 6.6 Graphing Linear Equations

What You Will Learn Upon completion of this section, you will be able to: Graph linear equations by plotting points. Graph linear equations by using the x- and y-intercepts. Determine the slope of a line. Graph linear equations by using the slope and y-intercept.

Cartesian Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis. The point of intersection is the origin.

Plotting Points – Ordered Pairs Each point in the xy-plane corresponds to a unique ordered pair (a, b). Plot the point (5, 3). Starting from the origin: Move 5 units right Move 3 units up

Example 2: A Parallelogram The points, A, B, and C are three vertices of a parallelogram with two sides parallel to the x-axis. Plot the three points below and determine the coordinates of the fourth vertex, D. A(1, 2) B(2, 4) C(7, 4)

Example 2: A Parallelogram Solution A parallelogram is a figure that has opposite sides that are of equal length and are parallel. The horizontal distance between points B and C is 5 units. Therefore, the horizontal distance between points A and D must also be 5 units.

Example 2: A Parallelogram Solution This problem has two possible solutions.

Example 2: A Parallelogram Solution

Graphing Linear Equations by Plotting Points A graph is an illustration of all the points whose coordinates satisfy an equation. All equations of the form ax + by = c, a ≠ 0, and b ≠ 0, will be straight lines when graphed. Thus, such equations are called linear equations in two variables.

Graphing Linear Equations by Plotting Points Since only two points are needed to draw a line, only two points are needed to graph a linear equation. It is always a good idea to plot a third point as a checkpoint. If no error has been made, all three points will be in a line, or collinear.

To Graph Linear Equations by Plotting Points 1. Solve the equation for y. 2. Select at least three values for x and find their corresponding values of y. 3. Plot the points. 4. The points should be in a straight line. Draw a line through the set of points and place arrow tips at both ends of the line.

Example 3: Graphing an Equation by Plotting Points Graph y = 2x + 2. Solution

Graphing by Using Intercepts The point where the line crosses the x-axis is called the x-intercept. The point where the line crosses the y-axis is called the y-intercept.

Finding the x- and y-Intercepts To find the x-intercept, set y = 0 and solve the equation for x. To find the y-intercept, set x = 0 and solve the equation for y.

Example 4: Graphing an Equation by Using Intercepts Graph 2x + 3y = 6 by using the x- and y-intercepts. Solution To find the x-intercept, set y = 0 and solve for x. The x-intercept is (3, 0).

Example 4: Graphing an Equation by Using Intercepts Solution To find the y-intercept, set x = 0 and solve for y. The y-intercept is (0, 2).

Example 4: Graphing an Equation by Using Intercepts Solution As a checkpoint, substitute x=-3 and determine the value of y. The checkpoint is (-3,4).

Example 4: Graphing an Equation by Using Intercepts Solution

Slope The slope is a measure of the “steepness” of a line. The slope of a line is a ratio of the vertical change to the horizontal change for any two points on the line.

Slope

Types of Slopes Positive slope rises from left to right.

Types of Slopes Negative slope falls from left to right.

Types of Slopes A horizontal line, which neither rises nor falls, has a slope of zero.

Types of Slopes The slope of a vertical line is undefined.

Example 5: Determining the Slope of a Line Find the slope of the line through the points (–1, –3) and (1, 5). Solution

Example 5: Determining the Slope of a Line Solution The slope of 4 means that there is a vertical change of 4 units for each horizontal change of 1 unit. The slope is positive, and the line rises from left to right.

Example 5: Determining the Slope of a Line Solution

Slope-Intercept Form of the Equation of a Line y = mx + b where m is the slope of the line and (0, b) is the y-intercept of the line.

To Graph Linear Equations by Using the Slope and y-Intercept 1. Solve the equation for y to place the equation in slope-intercept form. 2. Determine the slope and y-intercept from the equation.

To Graph Linear Equations by Using the Slope and y-Intercept 3. Plot the y-intercept. 4. Obtain a second point using the slope. 5. Draw a straight line through the points.

Example 7: Writing an Equation in Slope-Intercept Form a) Write 4x – 3y = 9 in slope-intercept form. b) Graph the equation. Solution

Example 7: Writing an Equation in Slope-Intercept Form Solution The y-intercept is (0, –3) and the slope is 4/3.

Example 8: Determining the Equation of a Line from Its Graph Determine the equation of the line. Solution The y-intercept is (0, 1), so b = 1. The change in y is 1 unit for every 3-unit change in x. Thus, m, the slope of the line, is

Example 8: Determining the Equation of a Line from Its Graph Solution The equation of the line is

Example 9: Horizontal and Vertical Lines In the Cartesian coordinate system, graph (a) y = 2 and (b) x = –3. Solution For any value if x, the value of y is 2.

Example 9: Horizontal and Vertical Lines Solution For any value of y, the value of x is –3.

Dependent Variable and Independent Variables In graphing the equations in this section, we labeled the horizontal axis the x-axis and the vertical axis the y-axis. For each equation, we can determine values for y by substituting values for x.

Dependent Variable and Independent Variables Since the value of y depends on the value of x, we refer to y as the dependent variable and x as the independent variable. We label the vertical axis with the dependent variable and the horizontal axis with the independent variable.