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Meaning of Slope for Equations, Graphs, and Tables

Section 1.4. Meaning of Slope for Equations, Graphs, and Tables. Section 1.4. Slide 2. Finding Slope from a Linear Equation. Finding Slope from a Linear Equation. Example. Find the slope of the line. Solution. x y 0 1 3 5 3 7.

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Meaning of Slope for Equations, Graphs, and Tables

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  1. Section 1.4 Meaning of Slope for Equations, Graphs, and Tables

  2. Section 1.4 Slide 2 Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution x y 0 1 3 5 3 7 Create a table using x = 1, 2, 3. Then sketch the graph.

  3. Section 1.4 Slide 3 Finding Slope from a Linear Equation Finding Slope from a Linear Equation Observations Note the following three observations about the slope of the line The coefficient of x is 2, which is the slope. If the run is 1, then the rise is 2. As the value of x increases by 1, the value of y increases by 2.

  4. Section 1.4 Slide 4 Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution x y 0 8 5 2 3 –1 Create a table using x = 1, 2, 3. Then sketch the graph.

  5. Section 1.4 Slide 5 Finding Slope from a Linear Equation Finding Slope from a Linear Equation Property For a linear equation of the form , m is the slope of the line. Example Are the lines parallel, perpendicular, or neither?

  6. Section 1.4 Slide 6 Finding Slope from a Linear Equation Finding Slope from a Linear Equation Property For the line the slope is For the other equation we solve for y: Original Equation Add 10x to both sides. Combine & rearrange terms Divide both sides by 12. Simplify.

  7. Section 1.4 Slide 7 Finding Slope from a Linear Equation Finding Slope from a Linear Equation Solution Continued For the line the slope is Since the slopes are the same for both equations, the lines are parallel Graphing Calculator We use ZStandard followed by ZSquare to draw the line in the same coordinate system.

  8. Section 1.4 Slide 8 Vertical Change Property Vertical Change Property Property For the line , if the run is 1, then the rise is m. Vertical Change property for a positive slope. Vertical Change property for a negative slope.

  9. Section 1.4 Slide 9 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 Property For a linear equation of the form , the y-intercept is (0, b).

  10. Section 1.4 Slide 10 Finding the y-intercept of a Linear Line Finding the y-Intercept of linear Equation Example What is the y-intercept of Solution • b is equal to 3, so the y-intercept is (0, 3) Definition If an equation of the form , we say that it is in slope-intercept form.

  11. Section 1.4 Slide 11 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 To graph: Plot the y-intercept, (0, 1). (continued)

  12. Section 1.4 Slide 12 Graphing Linear Equations Graphing Linear Equations Solution Continued • 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). • Sketch the line that contains these two points.

  13. Section 1.4 Slide 13 Graphing Linear Equations Graphing Linear Equations Guidelines • To sketch the graph of a linear equation of the form • Plot the y-intercept (0, b). • Use m = to plot a second point. • Sketch the line that passes through the two plotted points.

  14. Section 1.4 Slide 14 Graphing Linear Equations Graphing Linear Equations Example Sketch the graph of 2x + 3y = 6. Solution First we rewrite into slope-intercept form: Original Equation Subtract 2x from both sides. Combine & rearrange terms Divide both sides by 3.

  15. Section 1.4 Slide 15 Graphing Linear Equations Graphing Linear Equations Solution Continued • y-intercept: (0, 2) Slope: • 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).

  16. Section 1.4 Slide 16 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.

  17. Section 1.4 Slide 17 Graphing Linear Equations Graphing Linear Equations Example Determine the slope and the y-intercept of ax + by=c, where a, b, and c are constants and b is nonzero. 2. Find the slope and the y-intercept of the graph of 3x + 7y = 5. Solution First we rewrite into slope-intercept form:

  18. Section 1.4 Slide 18 Graphing Linear Equations Graphing Linear Equations Solution Continued Slope is and the y-intercept is

  19. Section 1.4 Slide 19 Graphing Linear Equations Graphing Linear Equations Solution Continued Given that ax + by = c in slope-intercept form is . • , then given 3x + 7y = 5, we substitute . 3 for a,7 for b and 5 for c. Thus, the slope, .

  20. Section 1.4 Slide 20 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.

  21. Section 1.4 Slide 21 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.

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