1. Chapter 4 Danica Reardon, Caroline Song, and Stephanie Knill

2. 4.1- Using an Equation to Draw a Graph You can use the equation y=mx+b to draw graphs m is meaning the slope of the line b is the y-intercept y-intercept slope

3. 4.1 – Part One the slope is 2 y-intercept is 5 Start by locating the y-intercept and marking it Since slope = rise over run, and 2 = 2/1, count 2 spaces up and one to the right Draw the line it creates

4. 4.1 – Part Two There are other ways of graphing the line One way is to take the coordinates and make a table From this, you can also get the equation xy -3 -21 3 05 17 29 311

5. 4.1 – Part Three Another way is by mapping Draw 2 ovals In one put the x coordinates In the other the y coordinates Draw arrows to connect them -3 -2 0 1 2 3 1 3 5 7 9 11 x y

6. Using A Graphing Calculator

7. Drawing Graphs Drawing graphs on these calculators are simple Click the button and type in your slope and y-intercept Make sure the plot buttons are not shaded in like below. If yes, go to it and press

8. Drawing Graphs Cont’d Press the button You should get something like

9. Stat Plots Can also use your tables Press then click In put your x coordinates In put your y-coordinates

10. Stat Plots Part 2 Press and click on Now press graph, you should get something like that

11. 4.2- The Slope of a Line Constant Slope Property: The slopes of all line segments of a line are equal

12. As you can tell, the slopes of all of these lines is ½. Therefore we can conclude that every segment on this line has a slope of ½, thus proving the statement we first made Proof Calculate the slopes of each line segment: Slope of AB Slope of BC Slope of CD Equation: x-2y+4=0 1 1 1 2 2 2 A(-4,0) B(-2,1) C(0,2) D(2,3)

13. Relationship Between Slopes of Parallel Line Segments If the slopes of two lines are equal, the lines are parallel Therefore, if two non-vertical lines are parallel, their slopes are equal Line AB: Line CD: C(0,-2) B(0,2) D(2,0) A(-2,0)

14. Relationship Between Slopes of Perpendicular Line Segments If the slopes of two lines are negative reciprocals, the lines are perpendicular Therefore, if two lines are perpendicular (and neither are vertical), their slopes are negative reciprocals

15. Relationship Between Slopes of Perpendicular Line Segments Proof: Line AB: Since parallel lines are equal, line CD also has a slope of 1 Line AC: Since parallel lines are equal, line BD also has a slope of –1 As you can see slope of AB/CD is 1 and the negative reciprocal of this is -1 which is the slope of AC/BD; This proves the statement we first made C(0,-2) B(0,2) D(2,0) A(-2,0)

16. 4.3 The Equation of a Line: Part 1

17. Graphing an Equation: y=3x+2 Method 1: 1. Draw table of values 2. Plot points on grid 3. Join points with a straight line xy -2-4 02 15 A(-2,-4) D(1,5) C(0,2) B(-1,-1) By Increasing x by 1 It increases y by 3

18. Graphing an Equation: y=3x+2 Method 2: There is also another method for graphing linear equations that does not need a table of values This method is based on two different numbers in the equation y=mx+b: The Slope This is the coefficient of x For y=3x+2, the slope is 3 The equation y=mx+b is a straight line where m represents slope and b represents y- intercept The y-intercept This is the value of b, and the value of y when x=o For y=3x+2, the slope is 3

19. 4.4- The Equation of a Line: Part 2 In the last section, we got the equation of a line when it’s y-intercept and slope is known There are 4 cases you should be able to solve

20. 4.4 – Case 1 Let’s find the equation of a line with the coordinates(1,2) and (3,6)

21. 4.4 – Case 1 first find the slope

22. 4.4 – Case 1 Remember the formula Plugging in your first coordinate, you get the formula y axis x axis slope Unknown y-intercept

23. 4.4 – Case 1 Now, solve for b Go back to the formula

24. 4.4 – Case 1 This time, do not enter anything for y or x Instead, insert your slope and y-intercept You get the equation

25. 4.4 – Case 2 Now, let’s find the equation of a line with the slope of –1 and a y-intercept of 4

26. 4.4 – Case 2 You have all the information necessary to make the equation So,

27. 4.4 – Case 3 Solving equations with 1 coordinate and the slope (1,5) and a slope of

28. 4.4 – Case 3 Put it into the equation like you did in Case 1 and solve for b So

29. 4.4 – Case 4 Slope of 5 x-intercept of 3

30. 4.4 – Case 4 At x-intercept y=0 Same as Case 2 Solve it the same

31. 4.4 – Case 4

32. 4.5- Interpreting the Equation Ax+By+C=0 Determining the intercepts 1Determining the intercepts 1 -To find the y-intercept, substitute x for 0. ex. 2x+4y-8=0 2(0)+4y-8=0 4y=8 y=4

33. 4.5- Interpreting the Equation Ax+By+C=0 The Standard Form:The Standard Form: Ax+By+C=0 Ax+By+C=0 The Slope-Intercept Form:The Slope-Intercept Form: y=mx+b Slope y-intercept

34. 4.5- Interpreting the Equation Ax+By+C=0 Determining the intercepts 2Determining the intercepts 2 -To find the x-intercept, substitute y for 0. ex. 2x+4y-8=0 2x+4(0)-8=0 2x=8 x=4

35. 4.5- Interpreting the Equation Ax+By+C=0 Solving the equation for ySolving the equation for y -In order to find the slope and y-intercept, you need to convert the standard form to the slope-intercept form. ex. 2x+4y-8=0 4y-8=-2x 4y=-2x+8 y=-0.5x+2 y-intercept Slope

36. 4.5- Interpreting the Equation Ax+By+C=0 Formulas for the slope, y-intercept,Formulas for the slope, y-intercept, and x-intercept Ax+By+C=0 Therefore: By+C=-Ax *Slope: -A/B By=-Ax-C *y-intercept: - C/B y=-A/Bx-C/B *x-intercept: -C/A This way, you just need to substitute the A, B and C values to find the slope, y-intercept and x- intercept.

38. Bibliography Internet Sources: http://en.wikipedia.org/wiki/Y%3Dmx%2Bb “Linear Equation” Wikipedia, 2007http://en.wikipedia.org/wiki/Y%3Dmx%2Bb www.tea.state.tx.us/.../images/47graphicaa.gif “Grade 10 Math Online Test” Texas Education Agency, 2006www.tea.state.tx.us/.../images/47graphicaa.gif Book Sources: Alexander, Robert & Kelly, Brendan. Mathematics 10. Addison Welsey Longman Ltd. 1998

39. For all Math 10H students… Click on this link : http://www.youtube.com/watch? v=jF-yU7Rq2XQ http://www.youtube.com/watch? v=jF-yU7Rq2XQ