1. C ollege A lgebra Linear and Quadratic Functions (Chapter2) 1

2. 2 Objectives Chapter2 Cover the topics in ( Section 2.1:Linear functions) After completing these sections, you should be able to: 1.Calculate and Interpret the Slope of a Line 2. Graph Lines Given a Point and the Slope 3. Use the Point-Slope Form of a Line 4. Find the Equation of a Line Given Two Points 5. Write the Equation of a Line in Slope-Intercept From and in General Form.

3. 3 Slope of a Line Chapter2 Let and be two distinct points with. The slope m of the non- vertical line L containing P and Q is defined by the formula If, L is a vertical line and the slope m of L is undefined (since this results in division by 0).

4. 4 Slope of a Line Chapter2 y x Slope can be though of as the ratio of the vertical change ( ) to the horizontal change ( (, often termed “ rise over run ”.

5. 5 Slope of a Line Chapter2 Definition of Slope The slope of the line through the distinct points (x 1, y 1 ) and (x 2, y 2 ) is where x 2 – x 1 = 0. Definition of Slope The slope of the line through the distinct points (x 1, y 1 ) and (x 2, y 2 ) is where x 2 – x 1 = 0. Change in y Change in x = Rise Run = y 2 – y 1 x 2 – x 1 (x 1, y 1 ) x1x1 y1y1 x2x2 y2y2 (x 2, y 2 ) Rise y 2 – y 1 Run x 2 – x 1 x y Definition of Slope

6. 6 Slope of a Line Chapter2 The Possibilities for a Line’s Slope Positive Slope x y m > 0 Line rises from left to right. Zero Slope x y m = 0 Line is horizontal. m is undefined Undefined Slope x y Line is vertical. Negative Slope x y m < 0 Line falls from left to right.

7. 7 Slope of a Line Chapter2 L y x If, then is zero and the slope is undefined. Plotting the two points results in the graph of a vertical line with the equation.

8. 8 Slope of a Line Chapter2 Example(1): Find the slope of the line joining the points (3,8) and (-1,2).

9. 9 Slope of a Line Chapter2 Example(2): Draw the graph of the line passing through (1,4) with a slope of -3/2. Step 1: Plot the given point. Step 2: Use the slope to find another point on the line (vertical change = -3, horizontal change = 2). y x ( 1,4 ) 2 -3 (3,1)

10. 10 Slope of a Line Chapter2 Example(3): Draw the graph of the equation x = 2. y x x = 2

11. 11 Equation of a Line Chapter2

12. When writing an equation of a line, keep in mind that you ALWAYS need two pieces of information when you go to write an equation: 1.ANY point on the line 2.Slope Equation of a Line Chapter2 Point-Slope Form of an Equation of a Line An equation of a non-vertical line of slope m that passes through the point (x 1, y 1 ) is:

13. 13 Equation of a Line Chapter2 Example(4): Find an equation of a line with slope -2 passing through (-1,5).

14. 14 Equation of a Line Chapter2

15. 15 Equation of a Line Chapter2 Slope/Intercept Equation of a Line In this form, m represents the slope and b represents the y-intercept of the line.

16. 16 Equation of a Line Chapter2

17. 17 Equation of a Line Chapter2 Equation of a Horizontal Line A horizontal line is given by an equation of the form y = b where b is the y-intercept.

18. 18 Equation of a Line Chapter2 Equation of a Vertical Line A vertical line is given by an equation of the form x = a where a is the x-intercept.

19. 19 Equation of a Line Chapter2

20. 20 Equation of a Line Chapter2

21. 21 Equation of a Line Chapter2

22. 22 Equation of a Line Chapter2

23. 23 Equation of a Line Chapter2 The equation of a line L is in general form with it is written as where A, B, and C are three real numbers and A and B are not both 0. The equation of a line L is in slope-intercept form with it is written as y = mx + b where m is the slope of the line and (0,b) is the y-intercept.

24. 24 Equation of a Line Chapter2 Example: Find the slope m and y-intercept (0,b) of the graph of the line 3x - 2y + 6 = 0.

25. 25 Equation of a Line Chapter2 Example: Write the point-slope form of the equation of the line passing through (-1,3) with a slope of 4. Then solve the equation for y. Solution We use the point-slope equation of a line with m = 4, x 1 = -1, and y 1 = 3. This is the point-slope form of the equation. y – y 1 = m(x – x 1 ) Substitute the given values. y – 3 = 4[x – (-1)] We now have the point-slope form of the equation for the given line. y – 3 = 4(x + 1) We can solve the equation for y by applying the distributive property. y – 3 = 4x + 4 y = 4x + 7 Add 3 to both sides.

26. 26 Equation of a Line Chapter2 Example: Graph the line whose equation is y = 2/3 x + 2. Solution The equation of the line is in the form y = mx + b. We can find the slope, m, by identifying the coefficient of x. We can find the y-intercept, b, by identifying the constant term. y = 2/3 x + 2 The slope is 2/3. The y-intercept is 2. We plot the second point on the line by starting at (0, 2), the first point. Then move 2 units up (the rise) and 3 units to the right (the run). This gives us a second point at (3, 4). -5-4-3-212345 5 4 3 2 1 -2 -3 -4 -5 We need two points in order to graph the line. We can use the y- intercept, 2, to obtain the first point (0, 2). Plot this point on the y-axis.

27. 27 Equation of a Line Chapter2 Equation of a Horizontal Line A horizontal line is given by an equation of the form y = b where b is the y-intercept. Equation of a Vertical Line A vertical line is given by an equation of the form x = a where a is the x-intercept.

28. 28 Equation of a Line Chapter2

29. 29 Equation of a Line Chapter2 Equations of Lines Point-slope form: y – y 1 = m(x – x 1 ) Slope-intercept form: y = m x + b Horizontal line: y = b Vertical line: x = a General form: Ax + By + C = 0

30. 30 Parallel and Perpendicular Lines Chapter2 Definitions: Parallel Lines Two lines are said to be parallel if they do not have any points in common. Two distinct non-vertical lines are parallel if and only if they have the same slope and have different y-intercepts.

31. 31 Parallel and Perpendicular Lines Chapter2

32. 32 Parallel and Perpendicular Lines Chapter2 Definitions: Perpendicular Lines Two lines are said to be perpendicular if they intersect at a right angle. Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.

33. 33 Parallel and Perpendicular Lines Chapter2

34. 34 Parallel and Perpendicular Lines Chapter2 Example: Find the equation of the line parallel to y = -3x + 5 passing through (1,5). Since parallel lines have the same slope, the slope of the parallel line is m = -3.

35. 35 Parallel and Perpendicular Lines Chapter2 Example: Find the equation of the line perpendicular to y = -3x + 5 passing through (1,5). Slope of perpendicular line:

36. 36 Parallel and Perpendicular Lines Chapter2 Slope and Parallel Lines If two non-vertical lines are parallel, then they have the same slope. If two distinct non-vertical lines have the same slope, then they are parallel. Two distinct vertical lines, both with undefined slopes, are parallel.

37. 37 Parallel and Perpendicular Lines Chapter2 Example: Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form. Solution We are looking for the equation of the line shown on the left. Notice that the line passes through the point (-3, 2). Using the point-slope form of the line’s equation, we have x 1 = -3 and y 1 = 2. y = 2x + 1 -5-4-3-212345 5 4 3 2 1 -2 -3 -4 -5 (-3, 2) Rise = 2 Run = 1 y – y 1 = m(x – x 1 ) y 1 = 2x 1 = -3

38. 38 Parallel and Perpendicular Lines Chapter2 Solution Parallel lines have the same slope. Because the slope of the given line is 2, m = 2 for the new equation. -5-4-3-212345 5 4 3 2 1 -2 -3 -4 -5 (-3, 2) Rise = 2 Run = 1 y = 2x + 1 y – y 1 = m(x – x 1 ) y 1 = 2m = 2x 1 = -3 Example cont.

39. 39 Parallel and Perpendicular Lines Chapter2 Solution The point-slope form of the line’s equation is y – 2 = 2[x – (-3)] y – 2 = 2(x + 3) Solving for y, we obtain the slope-intercept form of the equation. y – 2 = 2x + 6 Apply the distributive property. y = 2x + 8 Add 2 to both sides. This is the slope-intercept form of the equation. Example cont.

40. 40 Parallel and Perpendicular Lines Chapter2 Slope and Perpendicular Lines If two non-vertical lines are perpendicular, then the product of their slopes is –1. If the product of the slopes of two lines is –1, then the lines are perpendicular. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. Slope and Perpendicular Lines If two non-vertical lines are perpendicular, then the product of their slopes is –1. If the product of the slopes of two lines is –1, then the lines are perpendicular. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines. 90°

41. 41 Parallel and Perpendicular Lines Chapter2 Example: Find the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0. Solution We begin by writing the equation of the given line in slope- intercept form. Solve for y. 2x + 4y – 4 = 0 This is the given equation. 4y = -2x + 4 To isolate the y-term, subtract x and add 4 on both sides. Slope is –1/2. y = -2/4x + 1 Divide both sides by 4. The given line has slope –1/2. Any line perpendicular to this line has a slope that is the negative reciprocal, 2.