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Graphing Linear Equations

Graphing Linear Equations. Linear Equation. An equation for which the graph is a line. Solution. Any ordered pair of numbers that makes a linear equation true. (9,0) IS ONE SOLUTION FOR Y = X - 9. Linear Equation. Example: y = x + 3. Graphing. Step 1: ~ Three Point Method ~

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Graphing Linear Equations

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  1. Graphing Linear Equations

  2. Linear Equation An equation for which the graph is a line

  3. Solution Any ordered pair of numbers that makes a linear equation true. (9,0) IS ONE SOLUTION FOR Y = X - 9

  4. Linear Equation Example: y = x + 3

  5. Graphing Step 1: ~ Three Point Method ~ Choose 3 values for x

  6. Graphing Step 2: Find solutions using table y = x + 3 Y | X 0 1 2

  7. Graphing Step 3: Graph the points from the table (0,3) (1,4) (2,5)

  8. Graphing Step 4: Draw a line to connect them

  9. Try These • Graph using a table (3 point method) 1) y = x + 3 2) y = x - 4

  10. X-intercept Where the line crosses the x-axis

  11. X-intercept The x-intercept has a y coordinate of ZERO

  12. X-intercept To find the x-intercept, plug in ZERO for y and solve

  13. Slope Describes the steepness of a line

  14. Slope Equal to: Rise Run

  15. Rise The change vertically, the change in y

  16. Run The change horizontally or the change in x

  17. Finding Slope Step 1: Find 2 points on a line (2, 3) (5, 4) (x1, y1) (x2, y2)

  18. Finding Slope Step 2: Find the RISE between these 2 points Y2 - Y1 = 4 - 3 = 1

  19. Finding Slope Step 3: Find the RUN between these 2 points X2 - X1 = 5 - 2 = 3

  20. Finding Slope Step 4: Write the RISE over RUN as a ratio Y2 - Y1= 1 X2 - X1 3

  21. Y-intercept Where the line crosses the y-axis

  22. Y-intercept The y-intercept has an x-coordinate of ZERO

  23. Y-intercept To find the y-intercept, plug in ZERO for x and solve

  24. Slope-Intercept y = mx + b m = slope b = y-intercept

  25. Step 1: Mark a point on the y-intercept

  26. Step 2: Define slope as a fraction...

  27. Step 3: Numerator is the vertical change (RISE)

  28. Step 4: Denominator is the horizontal change (RUN)

  29. Step 5: Graph at least 3 points and connect the dots

  30. Graphing Quadratic Functions • Definitions • 3 forms for a quad. function • Steps for graphing each form • Examples • Changing between eqn. forms

  31. Quadratic Function • A function of the form y=ax2+bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation:

  32. Vertex- • The lowest or highest point of a parabola. Vertex Axis of symmetry- • The vertical line through the vertex of the parabola. Axis of Symmetry

  33. Standard Form Equation y=ax2 + bx + c • If a is positive, u opens up If a is negative, u opens down • The x-coordinate of the vertex is at • To find the y-coordinate of the vertex, plug the x-coordinate into the given eqn. • The axis of symmetry is the vertical line x= • Choose 2 x-values on either side of the vertex x-coordinate. Use the eqn to find the corresponding y-values. • Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

  34. Example 1: Graph y=2x2-8x+6 • Axis of symmetry is the vertical line x=2 • Table of values for other points: x y • 0 6 • 1 0 • 2 -2 • 3 0 • 4 6 • * Graph! • a=2 Since a is positive the parabola will open up. • Vertex: use b=-8 and a=2 Vertex is: (2,-2) x=2

  35. Now you try one!y=-x2+x+12* Open up or down?* Vertex?* Axis of symmetry?* Table of values with 5 points?

  36. (.5,12) (-1,10) (2,10) (-2,6) (3,6) X = .5

  37. Vertex Form Equation y=a(x-h)2+k • If a is positive, parabola opens up If a is negative, parabola opens down. • The vertex is the point (h,k). • The axis of symmetry is the vertical line x=h. • Don’t forget about 2 points on either side of the vertex! (5 points total!)

  38. Now you try one! y=2(x-1)2+3 • Open up or down? • Vertex? • Axis of symmetry? • Table of values with 5 points?

  39. Example 2: Graphy=-.5(x+3)2+4 • a is negative (a = -.5), so parabola opens down. • Vertex is (h,k) or (-3,4) • Axis of symmetry is the vertical line x = -3 • Table of values x y -1 2 -2 3.5 -3 4 -4 3.5 -5 2 Vertex (-3,4) (-4,3.5) (-2,3.5) (-5,2) (-1,2) x=-3

  40. (-1, 11) (3,11) X = 1 (0,5) (2,5) (1,3)

  41. Intercept Form Equation y=a(x-p)(x-q) • The x-intercepts are the points (p,0) and (q,0). • The axis of symmetry is the vertical line x= • The x-coordinate of the vertex is • To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. • If a is positive, parabola opens up If a is negative, parabola opens down.

  42. Example 3: Graph y=-(x+2)(x-4) • The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) • Since a is negative, parabola opens down. • The x-intercepts are (-2,0) and (4,0) • To find the x-coord. of the vertex, use • To find the y-coord., plug 1 in for x. • Vertex (1,9) (1,9) (-2,0) (4,0) x=1

  43. Now you try one! y=2(x-3)(x+1) • Open up or down? • X-intercepts? • Vertex? • Axis of symmetry?

  44. x=1 (-1,0) (3,0) (1,-8)

  45. Changing from vertex or intercepts form to standard form • The key is to FOIL! (first, outside, inside, last) • Ex: y=-(x+4)(x-9) Ex: y=3(x-1)2+8 =-(x2-9x+4x-36) =3(x-1)(x-1)+8 =-(x2-5x-36) =3(x2-x-x+1)+8 y=-x2+5x+36 =3(x2-2x+1)+8 =3x2-6x+3+8 y=3x2-6x+11

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