Chapter 3 Section 4

Writing and Graphing Equations of Lines 3.4 Writing and Graphing Equations of Lines Use the slope-intercept form of the equation of a line. Graph a line by using its slope and a point on the line. Write an equation of a line by using its slope and any point on the line. Write an equation of a line by using two points on the line. Write an equation of a line that fits a data set. 2 3 4 5

Use the slope-intercept form of the equation of a line. Objective 1 Use the slope-intercept form of the equation of a line. Slide 3.4-3

Use the slope-intercept form of the equation of line. In Section 3.3, we found the slope of a line by solving for y. In that form, the slope is the coefficient of x. For, example, the slope of the line with equation y = 2x + 3 is 2. So, what does 3 represent? Suppose a line has a slope m and y-intercept (0,b). We can find an equation of this line by choosing another point (x,y) on the line as shown. Then we use the slope formula. Change in y-values Change in x-values Subtract in the denominator. Multiply by x. Add b to both sides. Rewrite. Slide 3.4-4

Use the slope-intercept form of the equation of line. The result is the slope-intercept form of the equation of a line, because both the slope and the y-intercept of the line can be read directly from the equation. For the line with the equation y = 2x + 3, the number 3 gives the y-intercept (0,3). Slope-Intercept Form The slope-intercept form of the equation of a line with slope m and y-intercept (0,b) is Where m is the slope and b is the y-intercept (0,b). Slide 3.4-5

Identifying Slopes and y-Intercepts EXAMPLE 1 Identifying Slopes and y-Intercepts Identify the slope and y-intercept of the line with each equation. Solution: Slope: y-intercept: (0,− 6) Slope: − 1 y-intercept: (0,0) Slide 3.4-6

Writing an Equation of a Line EXAMPLE 2 Writing an Equation of a Line Write an equation of the line with slope −1 and y-intercept (0,5). Solution: Slide 3.4-7

Graph a line by using its slope and a point on the line. Objective 2 Graph a line by using its slope and a point on the line. Slide 3.4-8

Graph a line by using its slope and a point on the line. Graphing a Line by Using the Slope and y-Intercept Step 1: Write the equation in slope-intercept form, if necessary, by solving for y. Step 2: Identify the y-intercept. Graph the point (0,b). Step 3: Identify slope m of the line. Use the geometric interpretation of slope (“rise over run”) to find another point on the graph by counting from the y-intercept. Step 4: Join the two points with a line to obtain the graph. Slide 3.4-9

Graphing Lines by Using Slopes and y-Intercepts EXAMPLE 3 Graphing Lines by Using Slopes and y-Intercepts Graph 3x – 4y = 8 by using the slope and y-intercept. Solution: Slope intercept form Slide 3.4-10

Graphing a Line by Using the Slope and a Point EXAMPLE 4 Graphing a Line by Using the Slope and a Point Graph the line through (2,−3) with slope Solution: Make sure when you begin counting for a second point you begin at the given point, not at the origin. Slide 3.4-11

Objective 3 Write an equation of a line by using its slope and any point on the line. Slide 3.4-12

Write an equation of a line by using its slope and any point on the line. We can use the slope-intercept form to write the equation of a line if we know the slope and any point on the line. Slide 3.4-13

Using the Slope-Intercept Form to Write an Equation EXAMPLE 5 Using the Slope-Intercept Form to Write an Equation Write an equation, in slope-intercept form, of the line having slope −2 and passing through the point (−1,4). Solution: The slope-intercept form is Slide 3.4-14

Write an equation of a line by using its slope and any point on the line. There is another form that can be used to write the equation of a line. To develop this form, let m represent the slope of a line and let (x1,y1) represent a given point on the line. Let (x, y) represent any other point on the line. Definition of slope Multiply each side by x − x1. Rewrite. Point-Slope Form The point-slope form of the equation of a line with slope m passing through point (x1,y1) is Slope Given point Slide 3.4-15

Using the Point-Slope Form to Write Equations EXAMPLE 6 Using the Point-Slope Form to Write Equations Write an equation of the line through (5,2), with the slope Give the final answer in slope-intercept form. Solution: Slide 3.4-16

Write an equation of a line by using two points on the line. Objective 4 Write an equation of a line by using two points on the line. Slide 3.4-17

Write an equation of a line by using two points on the line. Many of the linear equations in Section 3.1−3.3 were given in the form called standard form, where A, B, and C are real numbers and A and B are not both 0. Slide 3.4-18

Writing the Equation of a Line by Using Two Points EXAMPLE 7 Writing the Equation of a Line by Using Two Points Find an equation of the line through the points (2,5) and (−1,6). Give the final answer in slope-intercept form and standard form. Solution: Standard form Slope-intercept form The same result would also be found by substituting the slope and either given point in slope-intercept form and then solving for b. Slide 3.4-19

Summary of the forms of linear equations. Slide 3.4-20

Write an equation of a line that fits a data set. Objective 5 Write an equation of a line that fits a data set. Slide 3.4-21

Writing an Equation of a Line That Describes Data EXAMPLE 8 Writing an Equation of a Line That Describes Data Use the points (3, 4645) and (7, 6185) to write an equation in slope-intercept form that approximates the data of the table. How well does this equation approximate the cost in 2005? Solution: The equation gives y = 5415 when x = 5, which is a very good approximation. Slide 3.4-22