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# Graphing Linear Equations

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1. Graphing Linear Equations • What we should learn • Graphing linear equations in one variable • Graphing linear equations in two variables • Graphing using intercepts • Slope • Graphing using slope • Solutions of linear equations • Graphs of absolute value • Solving absolute value equations.

2. 4.1 Graphing Linear Equations The graph of a point in two variables x and y is the ordered pair (x, y), where x and y are the coordinates of the point. x, the first number, tells how far to the left or right the point is from the vertical axis. y, the second number, tells how far up or down the point is from the vertical axis. For instance, the point (–1, 3) is the point that is 1 to left of vertical axis and is three up from horizontal axis.

3. 4.1 Graphing Linear Equations (x, y) coordinate system terms y Quadrant I Quadrant II (-3,2) 2 (3,2) x-axis x –3 1 2 3 (1,-2) (-3,-3) origin Quadrant III y-axis Quadrant IV

4. 4.1 Graphing Linear Equations The graph of an equation in two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation. For instance, the point (–1, 3) is on the graph of 2y – x = 7 because the equation is satisfied when –1 is substituted for x and 3 is substituted for y. That is, 2y – x = 7Original Equation 2(3) – (–1) = 7Substitute for x and y. 7 = 7Equation is satisfied.

5. 4.1 Graphing Linear Equations To sketch the graph of an equation, • Find several solution points of the equation by substituting various values for x and solving the equation for y. 2. Plot the points in the coordinate plane. • Connect the points using straight lines or smooth curves.

6. 4.1 Graphing Linear Equations Example: Sketch the graph of y = –2x + 3. 1. Find several solution points of the equation.

7. 4.1 Graphing Linear Equations Example: Sketch the graph of y = x - 1. 1. Find several solution points of the equation.

8. y This is the graph of the equation 2x + 3y = 12. (0,4) (6,0) x 2 -2 4.3 Quick Graphs Using Intercepts Equations of the form ax + by = c are called linear equations in two variables. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept.

9. y This is the graph of the equation 2x - 3y = 6. x (3,0) (0,-2) 4.3 Quick Graphs Using Intercepts Equations of the form ax + by = c are called linear equations in two variables. The point (0,-2) is the y-intercept. The point (3,0) is the x-intercept.

10. y 1 1 m = 2 4 x 2 -2 m = - 4.4 Slope m is undefined The slope of a line is a number, m, which measures its steepness. m = 2 m = 0

11. y2 – y1 , (x1 ≠ x2). m = x2 – x1 (x2, y2) y y2–y1 change in y (x1, y1) x2–x1 change in x x 4.4 Slope The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula The slope is the change in y divided by the change in x as we move along the line from (x1, y1) to (x2, y2).

12. y2 – y1 5 – 3 2 1 m = = = = x2 – x1 2 4 – 2 y x 4.4 Slope Example: Find the slope of the line passing through the points (2,3) and (4,5). Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5. (4, 5) 2 (2, 3) 2

13. y2 – y1 m = x2 – x1 y x 4.4 Slope Example: Find the slope of the line passing through the points (-1,2) and (3,1). Use the slope formula with x1= -1, y1 = 2, x2 = 3, and y2 = 1. (-1, 2) -1 (3, 1) 4

14. 4.5 Slope-Intercept Form A linear equation written in the form y = mx + b is in slope-intercept form. The slope is m and the y-intercept is (0, b). To graph an equation in slope-intercept form: 1. Write the equation in the form y = mx + b. Identify m and b. 2. Plot the y-intercept (0,b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope.

15. y x change in y 2 m = = 1 change in x (0,-4) (1, -2) 4.5 Slope-Intercept Form Example: Graph the line y = 2x– 4. • The equation y = 2x– 4 is in the slope-intercept form. So, m = 2 and b = -4. 2. Plot the y-intercept, (0,-4). 3. The slope is 2. 2 4. Start at the point (0,4). Count 1 unit to the right and 2 units up to locate a second point on the line. 1 The point (1,-2) is also on the line. 5. Draw the line through (0,4) and (1,-2).

16. 1 1 1 2 2 2 Example:The graph of the equation y – 3 = -(x – 4) is a line of slope m = - passing through the point (4,3). y m = - 8 (4, 3) 4 x 4 8 4.5 Slope-Intercept Form A linear equation written in the form y–y1 = m(x – x1) is in point-slope form. The graph of this equation is a line with slope m passing through the point (x1, y1).

17. Point-slope form y – y1 = m(x – x1) y – y1 = 3(x – x1) Let m = 3. y – 5 = 3(x – (-2)) Let (x1, y1)= (-2,5). y – 5 = 3(x + 2) Simplify. y = 3x + 11 Slope-intercept form 4.5 Slope-Intercept Form Example: Write the slope-intercept form for the equation of the line through the point (-2,5) with a slope of 3. Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1)= (-2,5).

18. m = Calculate the slope. 1 1 3 3 Point-slope form y – y1 = m(x – x1) 5 – 3 2 1 = - = - Use m = -and the point (4,3). -2 – 4 6 3 y – 3 = - (x – 4) 13 Slope-intercept form 3 y = - x + 4.5 Slope-Intercept Form Example: Write the slope-intercept form for the equation of the line through the points (4,3) and (-2,5).

19. 4.5 Slope-Intercept Form State the slope and y-intercept • y = -2x + 3 • x + y = 10 • 2x = 4 • y = 5 • x + 2y – 5 = 0 • y = x2 + 4

20. y (0, 4) y = 2x + 4 x y = 2x– 3 (0, -3) 4.5 Slope-Intercept Form Two lines are parallel if they have the same slope. If the lines have slopes m1 and m2, then the lines are parallel whenever m1 = m2. Example: The lines y = 2x – 3 and y = 2x + 4 have slopes m1 = 2 and m2 = 2. The lines are parallel.

21. y 1 1 1 1 m2= - 3 3 3 m1 x Example: The lines y = 3x – 1 and y = - x + 4 have slopes m1 = 3 and m2 = - . y = - x + 4 4.5 Slope-Intercept Form Two lines are perpendicular if their slopes are negative reciprocals of each other. If two lines have slopes m1 and m2, then the lines are perpendicular whenever y = 3x– 1 (0, 4) or m1m2 = -1. (0, -1) The lines are perpendicular.

22. 4.6 Solutions and x-Intercepts Use a Graphic Check of a Solution 1. Write the equation in the form ax + b = 0. 2. Sketch the graph of y = ax + b. 3. The solution of ax + b = 0 is the x-intercept of y = ax + b.

23. 4.6 Solutions and x-Intercepts add -4 to both sides Solve2x + 4 = -2 combine like terms 2x + 4 – 4 = -2 - 4 divide both sides by -2 2x = -6 x = -3 Get2x + 4 = -2 equal to 0 2x + 6 = 0 Graph y = 2x + 6

24. 4.6 Solutions and x-Intercepts Graph y = 2x + 6 x = -3 is the x-intercept and the solution to 2x + 6 = 0

25. 4.6 Solutions and x-Intercepts subtract 1 from both sides Solve3x + 1 = -5 combine like terms 3x + 1 – 1 = -5 - 1 divide both sides by 3 3x = -6 x = -2 Get3x + 1 = -5 equal to 0 3x + 6 = 0 Graph y = 3x + 6

26. 4.6 Solutions and x-Intercepts Graph y = 3x + 6 x = -2 is the x-intercept and the solution to 3x + 6 = 0

27. 4.7 Graph Absolute Value Graph y = |x|

28. 4.7 Graph Absolute Value Graph y = 2|x| - 1

29. 4.7 Graph Absolute Value Graph y = -|x| + 2

30. 4.8 Absolute Value Equations Let x be a variable of an algebraic expression and let a be real number such that a> 0. The solutions of the equation |x| = a are given by x = a and x = -a example|x| = 10 this is equivalent to the two equations x = 10 and x = -10

31. 4.8 Absolute Value Equations • Example solve |x + 3| = 5 • x + 3 = 5 or x + 3 = -5 • x = 2 or x = -8 • Solution set is {2, -8}

32. 4.8 Absolute Value Equations • Example solve |2x - 3| - 5 = 8 • |2x – 3| = 13 add 5 to both sides • 2x – 3 = 13 or 2x – 3 = -13 • 2x = 16 2x = -10 • x = 8 or x = -5 • The solution is { 8, -5}

33. 4.8 Absolute Value Equations • Example solve |3x - 1| - 4 = -2 • |3x – 1| = 2 add 4 to both sides • 3x – 1 = 2 or 3x – 1 = -2 • 3x = 3 3x = -1 • x = 1 or x = -1/3 • The solution is { -1/3, 1}