2. Chapter 4 – Coordinate Geometry: The Straight Line James Kim Michael Chang Math 10 Block : D

3. Table of Contents 4.1 – Using an Equation to Draw Graph 4.2 – The Slope of a Line 4.3 – The Equation of a Line :Part 1 4.4 – The Equation of a Line :Part 2 4.5 – Interpreting the Equation (Ax+By+C=0)

4. 4.1 - Using an Equation to Draw a Graph Equation of a Line Property The coordinates of every point on the line satisfy the equation of the line Every point whose coordinates satisfy the equation of the line is on the line

5. 4.1 - Using an Equation to Draw a Graph A basic equation y=mx+b m equals slope b equals y-intercept

6. 4.1 - Using an Equation to Draw a Graph Example (Using a calculator) Equation/ Y=2X+3 1. On a TI-83 Graphic calculator, go to Y= and type 2X+3 for Y 1 2. Press window, and type numbers. (X min=-10, X max=10, X sc1=1, Y min=-10, Y max=10, Y sc1=1 X res=1) 3. Press graph You would get this on your graph -

7. 4.1 - Using an Equation to Draw a Graph Example (No calculator) Equation/ Y=2X+3 1. You solve for Y, when X equals 1, 2, 3. You would get 5, 7, 9 for Y. 2. You draw zooms on the grid - - - 3. You draw line through the zooms - -

8. 4.2 - The Slope of a Line Constant Slope Property The Constant Slope Property allows us to define the slope of a line to be the slope of any segment of the line The Constant Slope Property is used to draw a line passing through a given point with a given slope If the slope of two lines are equal, the lines are parallel Conversely, if two non-vertical lines are parallel. Their slopes are equal. If the slopes of two lines are negative reciprocals, the lines are perpendicular Conversely, if two lines are perpendicular, their slopes are negative reciprocals

9. 4.2 - The Slope of a Line Example For this graph, the coordinates are given, which are (1,-1) and (-2,3) So the slope of this line segment is M = (3-(-1)) / (-2-1) The slope for this line is 4/-3

10. 4.3 - The Equation of a Line: Part 1 The graph of the equation y = mx+b is a straight line with slope m and y-intercept b Draw a graph with equation y=mx+b.

11. 4.3 – The Equation of a Line Part 1 Example Equation y=4x-3 The slope is 4 and y intercept is –3 In calculator, go to Y=, and put Y 1 = 4x-3 - - Then, press graph button and you will get - -

12. 4.4 - The Equation of a Line: Part 1 Ax + By + C = 0 Standard form of the equation of a line Collinear – Coordinates in the same straight line

13. 4.4 - The Equation of a Line: Part 1 4 Cases of solving the Equation of a line Case 1 : Given two coordinates Case 2 : Given slope and y-intercept Case 3 : One coordinate and the slope Case 4 : Slope and the x-intercept

14. Case #1 – Given two coordinates Example (4,3), (7,9) Find the slope 1. M = (9-3) / (7-4) = (6/3) = 2 2. Y = 2x + b 3. Choose 1 point to substitute 4. Y = 2x=b 3 = 2(4) + b b = -5 5. Y = 2x-5

15. Case #2 – Given slope and y -intercept Example 1. m = -2, y-int = or b=6 2. y = mx+b 3. y = -2x +6

16. Case #3 – Given one coordinate and the slope Example (x,y) = (3,-1), m=2 1. Y = 2x+b 2. –1= 6+b 3. –7=b 4. Y = 2x-7

17. Case #4 – Given slope and the x -intercept Example m = 5, x-int = 3 (3,0) 1. Y = 5x+b 2. 0 = 5(3)+b 3. –15 = b 4. Y= 5x-15

18. 4.5 - Interpreting the Equation Ax + By + C = 0 Determining the slope, x- intercept, and y-intercept Ax + By + C = 0 By = -Ax –C Y = -(-A/B) + (-C/B) Slope = (-A/B) Y intercept = (-C/B) Ax = -C X = (-C/A) X intercept = (-C/A)

19. 4.5 - Interpreting the Equation Ax + By + C = 0 Example Find slope, y-int, and x-int for this equation. Graph the equation 2x + 4y + 8 = 0 1. Slope : (-A/B) = (-2/4) = (-1/2) 2. Y-int : (-C/B) = (-8/4) = (-2) 3. X-int : (-C/A) = (-8/2) = (-4) 4. Equation : y = (-1/2)x -2 5. Graph : - - - - - - - - - - - - - - - -