1 / 47

Parabolas: Graphing Parabolas

Parabolas: Graphing Parabolas. Section 4.1. Definitions. Quadratic Equation is viewed as, a x 2 + b x + c = 0, where a ≠ 0 Parabola is an “u-shaped” graph If a is positive, it opens up If a is negative, it opens down Vertex is the highest or lowest point of the graph

selia
Télécharger la présentation

Parabolas: Graphing Parabolas

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Parabolas: Graphing Parabolas Section 4.1 4.1 - Graphing Parabolas

  2. 4.1 - Graphing Parabolas Definitions • Quadratic Equation is viewed as, ax2 + bx + c = 0, where a ≠ 0 • Parabola is an “u-shaped” graph • If a is positive, it opens up • If a is negative, it opens down • Vertex is the highest or lowest point of the graph • Axis of Symmetry is the vertical line passing though the vertex (x of vertex) represented by a DOTTED-line • Roots are also generally known as x-intercepts or solutions. It is where the graph crosses the x-axis

  3. 4.1 - Graphing Parabolas Definitions y = x2 – 4 X-intercept(s): Point(s) where the graph crosses the x-axis. Roots: The solutions to the equation . Ordered Pair: (–2, 0) Ordered Pair: (2, 0) Vertex: Minimum or maximum value Axis of Symmetry: Line that separates the graph in half; always is x equals

  4. 4.1 - Graphing Parabolas Equations • Quadratic equation: ax2 + bx + c = y • To find vertex:

  5. 4.1 - Graphing Parabolas Definitions • Quadratic Equation is viewed as, ax2 + bx + c = 0, where a ≠ 0 • Parent Functions of Parabola: y = x2 Change of a Change of c

  6. 4.1 - Graphing Parabolas The change in c • For shifts of c of ax2 + bx + c = y, the change of the c, the y-intercept shifts up or down

  7. 4.1 - Graphing Parabolas The change in a • For shifts of a of ax2 + bx + c = y, the bigger the a, the smaller the graph gets.

  8. 4.1 - Graphing Parabolas Steps • Make sure the parabola is in standard form, ax2 + bx + c = 0, where a ≠ 0 • Determine the vertex by using the equation, -b/2a • Determine the additional points by plugging in at least 4 additional points • Draw the arrowed graphs

  9. 4.1 - Graphing Parabolas Example 1 Determine the vertex of y = x2 – 4 and sketch the graph Is it in standard form? YES Determine the vertex.

  10. 4.1 - Graphing Parabolas Example 1 Determine the vertex of y = x2 – 4, sketch the graph, and describe the shift of the function Determine the vertex. To solve for x, plug into the equation, x = -b/2a

  11. 4.1 - Graphing Parabolas Example 1 Determine the vertex of y = x2 – 4, sketch the graph, and describe the shift of the function Determine the vertex. To solve for y of the vertex, plug the x into the equation

  12. 4.1 - Graphing Parabolas Definitions y = x2 – 4 X-intercept(s): Point(s) where the graph crosses the x-axis. Root(s): The solutions to the equation . Ordered Pair: (–2, 0) Ordered Pair: (2, 0) Vertex: Minimum or maximum value (0, –4) Axis of Symmetry: Line that separates the graph in half; always is x equals

  13. 4.1 - Graphing Parabolas Example 2 Determine the vertex of y = x2 – 9 and sketch the graph

  14. 4.1 - Graphing Parabolas Example 3 Given equation: y = x2 – 9 • Determine: • How it opens • Shift from the Parent Function • The Vertex • Roots • Y-Intercept • Axis of Symmetry • Domain • Range • Minimum/Maximum

  15. 4.1 - Graphing Parabolas Example 3 Given y = x2 – 9 • How it opens? • Is the leading coefficient positive or negative? POSITIVE OPENS UP

  16. 4.1 - Graphing Parabolas Example 3 From this equation: y = x2 – 9 • The shift from the parent function Moves 9 spots down

  17. 4.1 - Graphing Parabolas Example 3 • From this equation, identify the X-coordinate of the vertex y = x2 – 9 • Plug the equation in

  18. 4.1 - Graphing Parabolas Example 3 • From this equation, identify the Y-coordinate of the vertex y = x2 – 9 • Plug the equation in

  19. 4.1 - Graphing Parabolas Example 3 Given y = x2 – 9 • The vertex:

  20. 4.1 - Graphing Parabolas Example 3 Roots are where they cross the x-axis – Solve for x WHERE Y = 0

  21. 4.1 - Graphing Parabolas Example 3 From this equation: y = x2 – 9 • Y-intercept: y =(0, –9)

  22. 4.1 - Graphing Parabolas Example 3 Given y = x2 – 9 • Axis of Symmetry is the X of the vertex.

  23. 4.1 - Graphing Parabolas Example 3 • Domain is the listing of all of the x-coordinates ALL REAL NUMBERS (–∞, ∞)

  24. 4.1 - Graphing Parabolas Example 3 • Range is the listing of all of the y-coordinates y >-9 [–9, ∞) > > arrow u >

  25. 4.1 - Graphing Parabolas Your Turn • Given equation: y = x2 – 16 • Determine: • How it opens • Shift from the Parent Function • Roots • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum

  26. 4.1 - Graphing Parabolas Your Turn • Given equation, y = x2 – 16 • Determine: • How it opens • Shift from the Parent Function • Roots • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum up Down 16 (±4, 0) (0, -16) (0, -16) x = 0 All Reals y > -16 Min at -16

  27. 4.1 - Graphing Parabolas Example 3 • Given equation, y = 2x2 – 2 • Determine: • How it opens • Shift from the Parent Function • Roots • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum up Down 2 and compresses with a scale of 2 (1, 0) and (-1, 0) (0, -2) (0, -2) x = 0 All Reals y > -2 Min at -2

  28. 4.1 - Graphing Parabolas Example 4 • Given equation, y = 2x2 – 2x + 5 • Determine: • How it opens • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum up (0, 5) (0.5, 4.5) x = 0.5 All Real Numbers y > 4.5 Min at 4.5

  29. 4.1 - Graphing Parabolas Example 5 • Given equation: y = x2 + 6x + 9 • Roots • The Vertex

  30. 4.1 - Graphing Parabolas Example 4 • Given y = x2 + 6x + 9 • Roots: x = (–3, 0) Double Root

  31. 4.1 - Graphing Parabolas Example 4 • From this equation, identify the X-coordinate of the vertex y = x2 + 6x + 9 • Plug the equation in

  32. 4.1 - Graphing Parabolas Example 4 • From this equation, identify the Y-coordinate of the vertex y = x2 + 6x + 9 • Plug the equation in

  33. 4.1 - Graphing Parabolas Your Turn • Given equation: y = x2 + 8x + 16 • Roots • The Vertex

  34. 4.1 - Graphing Parabolas Quadratic Equation • Use the quadratic equation: f(x) = a(x – h)2 + k • To find vertex: • a:Reflection/Scaled stretch • h: Horizontal Translation • k: Vertical Translation (also known as ‘c’)

  35. 4.1 - Graphing Parabolas Example 5 • Given equation: y = (x + 4)2 – 9 • Determine: • How it opens • Shift from the Parent Function • Roots • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum

  36. 4.1 - Graphing Parabolas Example 5 • Given y = (x + 4)2 – 9 • How it opens? • Is the leading coefficient positive or negative? POSITIVE OPENS UP

  37. 4.1 - Graphing Parabolas Example 5 • Given y = (x + 4)2 – 9 • The shift from the parent function Down 9, left 4 spots

  38. 4.1 - Graphing Parabolas Example 5 • Given y = (x + 4)2 – 9 • Roots: x = (–7, 0) and (–1, 0)

  39. 4.1 - Graphing Parabolas Example 5 • Given y = (x + 4)2 – 9 • Y-intercept: y = (0, 7)

  40. 4.1 - Graphing Parabolas Example 5 • From this equation, identify the X-coordinate of the vertex y = (x + 4)2 – 9

  41. 4.1 - Graphing Parabolas Example 5 • Given: y = (x + 4)2 – 9 • The vertex:

  42. 4.1 - Graphing Parabolas Example 5 Given: y = (x + 4)2 – 9 Axis of Symmetry is the X of the vertex.

  43. 4.1 - Graphing Parabolas Example 5 • Domain is the listing of all of the x-coordinates ALL REAL NUMBERS

  44. 4.1 - Graphing Parabolas Example 5 • Range is the listing of all of the y-coordinates y >-9 arrow u >

  45. 4.1 - Graphing Parabolas Example 6 • Given equation: y = –(x – 3)2 + 5 • Determine: • How it opens • Shift from the Parent Function • Roots • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum down Right 3, Up 5 (0, -2) (0, -2) x = 0 All Reals y < -2 Min at -2

  46. 4.1 - Graphing Parabolas Your Turn • Given equation: y = 2(x + 3)2 - 6 • Determine: • How it opens • Shift from the Parent Function • Roots • Y-Intercept • The Vertex • Axis of Symmetry • Domain • Range • Minimum/Maximum up Left 3, Down 6, (0, -6) (-3, -6) x = -3 All Reals y > -6 Min at -6

  47. 4.1 - Graphing Parabolas Assignment WKST

More Related