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# Linear functions any_two_points

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### Linear functions any_two_points

1. 1. Linear Equations in Two Variables
2. 2. Linear Equations in Two Variables may be put in the form Ax + By = C, Where A, B, and C are real numbers and A and B are not both zero.
3. 3. Solutions to Linear Equations in Two Variables  Consider the equation  The equation’s solution set is infinite because there are an infinite number of x’s and y’s that make it TRUE.  For example, the ordered pair (0, 10) is a solution because  Can you list other ordered pairs that satisfy this equation? 5 2 20x y+ = ( ) ( )5 0 2 10 20+ = 5 2 20x y+ = Ordered Pairs are listed with the x-value first and the y-value second.
4. 4. Input-Output Machines  We can think of equations as input-output machines. The x-values being the “inputs” and the y-values being the “outputs.”  Choosing any value for input and plugging it into the equation, we solve for the output. y = -2x + 5 y = -2(4) + 5 y = -8 + 5 y = -3 x = 4 y = -3
5. 5. Using Tables to List Solutions  For an equation we can list some solutions in a table.  Or, we may list the solutions in ordered pairs . {(0,-4), (6,0), (3,-2), ( 3/2, -3), (-3,-6), (-6,-8), … } 2 3 12x y− = x y 0 -4 6 0 3 -2 3/2 -3 -3 -6 -6 -8 … …
6. 6. Graphing a Solution Set To obtain a more complete picture of a solution set we can graph the ordered pairs from our table onto a rectangular coordinate system. Let’s familiarize ourselves with the Cartesian coordinate system.
7. 7. Cartesian Plane x- axis y-axis Quadrant I (+,+) Quadrant II ( - ,+) Quadrant IV (+, - ) Quadrant III ( - , - ) origin
8. 8. Graphing Ordered Pairs on a Cartesian Plane x- axis y-axis 1) Begin at the origin 2) Use the x-coordinate to move right (+) or left (-) on the x-axis 3) From that position move either up(+) or down(-) according to the y- coordinate 4) Place a dot to indicate a point on the plane Examples: (0,-4) (6, 0) (-3,-6) (6,0) (0,-4) (-3, -6)
9. 9. Graphing More Ordered Pairs from our Table for the equation x y (3,-2) (3/2,-3) (-6, -8) 2 3 12x y− = •Plotting more points we see a pattern. •Connecting the points a line is formed. •We indicate that the pattern continues by placing arrows on the line. •Every point on this line is a solution of its equation.
10. 10. Graphing Linear Equations in Two Variables  The graph of any linear equation in two variables is a straight line.  Finding intercepts can be helpful when graphing.  The x-intercept is the point where the line crosses the x-axis.  The y-intercept is the point where the line crosses the y-axis. y x
11. 11. Graphing Linear Equations in Two Variables On our previous graph, y = 2x – 3y = 12, find the intercepts. The x-intercept is (6,0). The y-intercept is (0,-4). y x
12. 12. Finding INTERCEPTS To find theTo find the x-intercept: Plug inx-intercept: Plug in ZERO for y and solveZERO for y and solve for x.for x. 2x – 3y = 122x – 3y = 12 2x – 3(0) = 122x – 3(0) = 12 2x = 122x = 12 x = 6x = 6 Thus, the x-intercept is (6,0). To find the y-To find the y- intercept: Plug inintercept: Plug in ZERO for x and solveZERO for x and solve for y.for y. 2(0) – 3y = 122(0) – 3y = 12 2(0) – 3y = 122(0) – 3y = 12 -3y = 12-3y = 12 y = -4y = -4 Thus, the y-intercept isThus, the y-intercept is (0,-4).(0,-4).
13. 13. Special Lines y + 5 = 0 x = 3 y = -5 y x x y y = # is a horizontal line x = # is a vertical line
14. 14.  SLOPE- is the rate of change  We sometimes think of it as the steepness, slant, or grade. 2 1 2 1 y y y rise slope m x x x run ∆ − = = = = ∆ − Slope formula:
15. 15. Slope: Given 2 colinear points, find the slope. Find the slope of the line containing (3,2) and (-1,5). ( ) 2 1 2 1 2 5 3 3 1 4 y y m x x − − − = = = − − −
16. 16. Slopes Positive slopes rise from left to right Negative slopes fall from left to right
17. 17. Special Slopes  Vertical lines have UNDEFINED slope (run=0 --- undefined)  Horizontal lines have zero slope (rise = 0)  Parallel lines have the same slope (same slant)  Perpendicular lines have opposite reciprocal slopes 0m = 1 2 1 m m = − 1 2m m= m undefined=
18. 18. Read 2x: Bring out 1 whole sheet of paper: copy and answer With solution..
19. 19. Seatwork Find the slope of the following: 1. (3,2),(5,6) 2. (1,-2),(-2,0) 3. (4,6),(3,0) 4. (-9,6),(-10,3) 5. (-5,9),(-3,6) 6. (-4,8),(6,-1) 7. (0,2),(3,-2)