1. A4.a How Do I Interpret Slope of a Line As A Rate Of Change? Course 3 Warm Up Warm Up Lesson Presentation Lesson Presentation

2. Warm Up Evaluate each equation for x = –1, 0, and 1. 1. y = 3x 2. y = x – 7 3. y = 2x + 5 4. y = 6x – 2 –3, 0, 3 –8, –7, –6 3, 5, 7 Course 3 12-2 Slope of a Line –8, –2, 4

3. Learn to find the slope of a line and use slope to understand and draw graphs. Course 3 12-2 Slope of a Line

4. Linear equations have constant slope (constant rate of change). For a line on the coordinate plane, slope is the following ratio: vertical change horizontal change change in y change in x = This ratio is often referred to as, or “rise over run,” where rise indicates the number of units moved up or down (vertical) and run (horizontal) indicates the number of units moved to the left or right. Slope can be positive, negative, zero, or undefined. A line with positive slope goes up from left to right. A line with negative slope goes down from left to right. rise run Course 3 12-2 Slope of a Line

5. Course 3 12-2 Slope of a Line

6. Course 3 12-2 Slope of a Line

7. You can use any two points to find the slope of the line. Course 3 12-2 Slope of a Line Helpful Hint

8. If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is as follows: y2 – y1y2 – y1x2 – x1x2 – x1y2 – y1y2 – y1x2 – x1x2 – x1 Course 3 12-2 Slope of a Line

9. Find the slope of the line that passes through (–2, –3) and (4, 6). Example 1: Finding Slope, Given Two Points Let (x 1, y 1 ) be (–2, –3) and (x 2, y 2 ) be (4, 6). 6 – (–3) 4 – (–2) Substitute 6 for y 2, –3 for y 1, 4 for x 2, and –2 for x 1. 9 6 = The slope of the line that passes through (–2, –3) and (4, 6) is. 3 2 = y 2 – y 1 x 2 – x 1 3 2 = Course 3 12-2 Slope of a Line

10. Find the slope of the line that passes through (–4, –6) and (2, 3). Check It Out: Example 1 Let (x 1, y 1 ) be (–4, –6) and (x 2, y 2 ) be (2, 3). 3 – (–6) 2 – (–4) Substitute 3 for y 2, –6 for y 1, 2 for x 2, and –4 for x 1. 9 6 = The slope of the line that passes through (–4, –6) and (2, 3) is. 3 2 = y 2 – y 1 x 2 – x 1 3 2 = Course 3 12-2 Slope of a Line

11. Nonlinear equations have variable rates of change. This means that the rate of change is different between values. This is shown in a graph by a curved line. Course 3 12-2 Slope of a Line

12. Determine whether each graph shows a constant or variable rate of change. Explain your reasoning. Example 2A: Identifying Constant and Variable Rates of Change in Graphs Course 3 12-2 Slope of a Line The graph shows a constant rate of change. The slope between any two points is always the same.

13. Determine whether each graph shows a constant or variable rate of change. Explain your reasoning. Example 2B: Identifying Constant and Variable Rates of Change in Graphs Course 3 12-2 Slope of a Line The graph shows a variable rate of change. The slope between any two sets of points in Quadrant 1 is different.

14. Determine whether each graph shows a constant or variable rate of change. Explain your reasoning. Check It Out: Example 2A Course 3 12-2 Slope of a Line The graph shows a constant rate of change. The slope between any two points is always the same.

15. Determine whether each graph shows a constant or variable rate of change. Explain your reasoning. Check It Out: Example 2B Course 3 12-2 Slope of a Line The graph shows a variable rate of change. The slope is steeper at the ends than in the middle.

16. Example 3: Money Application The table shows the total cost of fruit per pound purchased at the grocery store. Use the data to make a graph. Find the slope of the line and explain what it shows. Course 3 12-2 Slope of a Line Graph the data. Pounds Cost Cost of Fruit

17. Additional Example 3 Continued Find the slope of the line: The slope of the line is 3. This means that for every pound of fruit, you will pay another $3. = y 3 – y 2 x 3 – x 2 15 5 = 30  15 10  5 = 3 Course 3 12-2 Slope of a Line

18. Check It Out: Example 3 The table shows the total cost of gas per gallon. Use the data to make a graph. Find the slope of the line and explain what it shows. Course 3 12-2 Slope of a Line Graph the data. Cost of Gas GallonsCost 00 36 612 6 9 9 6 0 3 3 x y Gallons Cost of Gas Cost

19. Check It Out: Example 3 Continued Find the slope of the line: The slope of the line is 2. This means that for every gallon of gas, you will pay another $2. = y 3 – y 2 x 3 – x 2 6 3 = 12  6 6  3 = 2 Course 3 12-2 Slope of a Line

20. Lesson Quiz: Part I Find the slope of the line passing through each pair of points. 1. (4, 3) and (–1, 1) 2. (–1, 5) and (4, 2) Insert Lesson Title Here 2 55 3 – Course 3 12-2 Slope of a Line

21. Lesson Quiz: Part II 3. The table shows how much money Susan earned as a house painter for one afternoon. Use the data to make a graph. Find the slope of the line and explain what it shows. Insert Lesson Title Here Course 3 12-2 Slope of a Line x y 64 2 8 101214 0 10 20 30 40 50 60 70 80 The slope of the line is 7. This means Susan earned $7 for each hour worked.