Section 1.4 Meaning of Slope for Equations, Graphs, and Tables
Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution Create a table using x = 1, 2, 3. Then sketch the graph. rise 2 m = = = 2 run 1 y x y 0 1 1 3 2 5 3 7 = 2x+ 1. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 2
Finding Slope from a Linear Equation Finding Slope from a Linear Equation Observations Note the following three observations about the slope of the line y = 2x+ 1. 1. The coefficient of x is 2, which is the slope. 2. If the run is 1, then the rise is 2. 3. As the value of x increases by 1, the value of y increases by 2. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 3
Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution Create a table using x = 1, 2, 3. Then sketch the graph. m rise 3 = = = 3 run 1 y = 3x+ 8. x y 0 8 1 5 2 2 3 1 Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 4
Vertical Change Property Vertical Change Property Property For the line m. y = mx + b, if the run is 1, then the rise is Vertical Change property for a positive slope. Vertical Change property for a negative slope. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 5
Finding the y-intercept of a Linear Line Finding the y - Intercept of linear Equation Sketching Equations: It s helpful to know the y-intercept. y-intercept has a x-value of 0. Substitute x = 0 gives y = m 0 + b= b Property ( ) For a linear equation of the form y = mx + b, the y- intercept is (0, b). Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 6
Finding the y-intercept of a Linear Line Finding the y - Intercept of linear Equation Example What is the y-intercept of y Solution 5 = x+ 3? 6 b is equal to 3, so the y-intercept is (0, 3) Definition If an equation of the form y = mx + b, we say that it is in slope-intercept form. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 7
Graphing Linear Equations Graphing Linear Equations Example Sketch the graph of y = 3x 1. Solution The y-intercept is (0, 1) and the slope is 3 rise 3 = = 1 run To graph: 1. Plot the y-intercept, (0, 1). (continued) Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 8
Graphing Linear Equations Graphing Linear Equations Solution Continued 2. From (0, 1), look 1 unit to the right and 3 units up to plot a second point, which we see by inspection is (1, 2). 3. Sketch the line that contains these two points. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 9
Graphing Linear Equations Graphing Linear Equations Guidelines To sketch the graph of a linear equation of the form y = mx + b 1. Plot the y-intercept (0, b). rise 2. Use m = to plot a second point. run 3. Sketch the line that passes through the two plotted points. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 10
Graphing Linear Equations Graphing Linear Equations Example Sketch the graph of 2x + 3y = 6. Solution First we rewrite into slope-intercept form: 2x+ 3y = 6 2x+ 3y 2x= 6 2x 3y = 2x+ 6 3y 2 6 = x + 3 3 3 Original Equation Subtract 2x from both sides. Combine & rearrange terms Divide both sides by 3. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 11
Graphing Linear Equations Graphing Linear Equations Solution Continued 2 -a a y = x+ 2 Simplfy: =- 3 b b 2 2 rise y-intercept: (0, 2) Slope: = = 3 3 run 1. Plot the y-intercept, (0, 2). 2. From the point (0, 2), look 3 units to the right and 2 units down to plot a second point, which we see by inspection is (3, 0). Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 12
Graphing Linear Equations Graphing Linear Equations Solution Continued 3. Then sketch the line that contains these two points. We can verify our result by checking that both (0, 2) and (3, 0) are solutions. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 13
Slope Addition Property Slope Addition Property Example For the following sets, is there a line that passes through them? If so, find the slope of that line. Solution Value of x increases by 1. Value of y changes by 3. The slope is 3. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 14
Slope Addition Property Slope Addition Property Solution Continued Set 2 Value of x increases by 1. Value of y changes by 5. So, the slope is 5. Set 3 Value of x increases by 1. Value of y does not change by the same value. Hence, not a line. Section 1.4 Lehmann, Intermediate Algebra, 4ed Slide 15