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Graphing Quadratic Functions

Learn to graph quadratic functions efficiently by understanding key concepts like vertex, symmetry, and minimum/maximum values. Practice examples included for better grasp of the topic.

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Graphing Quadratic Functions

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  1. Graphing Quadratic Functions Chapter 8

  2. Graphing f(x) = ax2 + bx + c I can graph quadratic functions of the form f(x) = ax2 + bx + c.

  3. Graphing f(x) = ax2 + bx + c Vocabulary (page 247 in Student Journal) quadratic function: a nonlinear function of the second degree • parabola: the u-shaped curve of the graph of a quadratic function

  4. Graphing f(x) = ax2 + bx + c • vertex: the lowest, or highest, point on a parabola, which is on the axis of symmetry axis of symmetry: the line that divides a parabola into 2 matching halves

  5. Graphing f(x) = ax2 + bx + c Vocabulary (page 252 in Student Journal) zero of a function: the value of x that makes y = 0 in a quadratic function, which is the place(s) where the parabola intersects the x-axis on the graph

  6. Graphing f(x) = ax2 + bx + c Vocabulary (page 257 in Student Journal) maximum value: the y-coordinate of the vertex when the parabola opens down (a < 0) minimum value: the y-coordinate of the vertex when the parabola opens up (a > 0)

  7. Graphing f(x) = ax2 + bx + c Core Concepts (pages 247 and 248 in Student Journal) Graphing f(x) = ax2 -when a > 1 the graph of f(x) = x2 is a vertical stretch -when 0 < a < 1 the graph of f(x) = x2 is a vertical shrink

  8. Graphing f(x) = ax2 + bx + c Graphing f(x) = ax2 (continued) -when a < -1 the graph of f(x) = x2 is a vertical stretch and is reflected in the x-axis -when -1 < a < 0 the graph of f(x) = x2 is a vertical shrink and is reflected in the x-axis

  9. Graphing f(x) = ax2 + bx + c Core Concepts (page 257 in Student Journal) Graphing f(x) = ax2 + bx + c -when a > 0, the graph opens up -when a < 0, the graph opens down -the y-intercept is c -the x-coordinate of the vertex is –b/2a

  10. Graphing f(x) = ax2 + bx + c Examples (pages 258 and 259 in Student Journal) Find the axis of symmetry and the vertex of the graph of the function. #1)

  11. Graphing f(x) = ax2 + bx + c Solution #1) x = 5, (5, -23)

  12. Graphing f(x) = ax2 + bx + c Graph the function. #5)

  13. Graphing f(x) = ax2 + bx + c Solution #5)

  14. Graphing f(x) = ax2 + bx + c Tell whether the function has a minimum or maximum value. Then find the value. #8)

  15. Graphing f(x) = ax2 + bx + c Solution #8) maximum, y = 14.5

  16. Graphing f(x) = ax2 + bx + c #14) The function represents the height of a rocket t seconds after its launched. The rocket explodes at its highest point. • When does the rocket explode? • At what height does the rocket explode?

  17. Graphing f(x) = ax2 + bx + c Solutions #14a) 7.8125 seconds #14b) 976.5625 feet

  18. Graphing f(x) = ax2 Additional Example (space on pages 257 and 258 in Student Journal) a) Identify the vertex, axis of symmetry, domain and range based on the graph.

  19. Graphing f(x) = ax2 Solution a) vertex at (2, -4), axis of symmetry is x = 2, domain is all real numbers, range is y greater than or equal to -4

  20. Graphing f(x) = a(x – h)2 + k I can graph quadratic functions in the form f(x) = a(x – h)2 + k.

  21. Graphing f(x) = a(x – h)2 + k Vocabulary (page 262 in Student Journal) • vertex form (of a quadratic function): a quadratic function in the form f(x) = a(x – h)2 + k

  22. Graphing f(x) = a(x – h)2 + k Core Concepts (pages 262 and 263 in Student Journal) Graphing f(x) = a(x – h)2 + k -a is the vertical stretch/shrink factor -h is the horizontal translation -k is the vertical translation -(h, k) are the coordinates of the vertex

  23. Graphing f(x) = a(x – h)2 + k • Examples(pages 263 and 264 in Student Journal) • Find the vertex and axis of symmetry of the graph of the function. • #5) • #8)

  24. Graphing f(x) = a(x – h)2 + k Solutions #5) (2, 0), x = 2 #8) (-1, -5), x = -1

  25. Graphing f(x) = a(x – h)2 + k Graph. #9)

  26. Graphing f(x) = a(x – h)2 + k Solution #9)

  27. Graphing f(x) = a(x – h)2 + k Additional Example (space on pages 262 and 263 in Student Journal) a) Write and graph a quadratic function that models the path of a stream of water from a fountain that has a maximum height of 4 feet at a time of 3 seconds and lands after 6 seconds.

  28. Graphing f(x) = a(x – h)2 + k Solution a) y = -4/9(x – 3)2 + 4

  29. Using Intercept Form I can graph quadratic functions of the form f(x) = a(x – p)(x – q).

  30. Using Intercept Form Vocabulary (page 267 in Student Journal) intercept form: a quadratic function written in factored, which is f(x) = a(x – p)(x – q)

  31. Using Intercept Form Core Concepts (page 267 in Student Journal) Graphing f(x) = a(x – p)(x – q) -x-intercepts are p and q -axis of symmetry is x = (p + q)/2 -a is the vertical stretch/shrink factor

  32. Using Intercept Form Examples(pages 268 and 269 in Student Journal) Find the x-intercepts and axis of symmetry of the graph of the function. #1)

  33. Using Intercept Form Solution #1) x-intercepts: -2, 4 axis of symmetry: x = 1

  34. Using Intercept Form Graph. #4)

  35. Using Intercept Form Solution #4)

  36. Using Intercept Form Find the zeros. #7)

  37. Using Intercept Form Solution #7) 1, -1

  38. Using Intercept Form Use zeros to graph the function. #9)

  39. Using Intercept Form Solution #9)

  40. Using Intercept Form Additional Examples (space on page 267 in Student Journal) Write a quadratic function in standard form that satisfies the given conditions. a) zeros at -1 and 5 b) vertex at (-3, -2)

  41. Using Intercept Form Possible Solutions a) y = x2 – 4x – 5 b) y = x2 + 6x + 7

  42. Comparing Linear, Exponential, and Quadratic Functions I can choose and write functions to model data.

  43. Comparing Linear, Exponential, and Quadratic Functions Vocabulary (page 272 in Student Journal) average rate of change: the slope of the line connecting 2 points on a graph

  44. Comparing Linear, Exponential, and Quadratic Functions Core Concepts (pages 272 and 273 in Student Journal) Linear, Exponential, and Quadratic Functions linear: y = mx + b exponential: y = abx quadratic: y = ax2 + bx + c

  45. Comparing Linear, Exponential, and Quadratic Functions Differences and Ratios of Functions linear: first differences of y-values are constant exponential: consecutive y-values have common ratio quadratic: second differences of y-values are constant

  46. Comparing Linear, Exponential, and Quadratic Functions Examples(pages 273 and 274 in Student Journal) Determine if the ordered pairs represent a linear, exponential or quadratic function. #1) (-3, 2), (-2, 4), (-4, 4), (-1, 8), (-5, 8) #2) (-3, 1), (-2, 2), (-1, 4), (0, 8), (2, 14)

  47. Comparing Linear, Exponential, and Quadratic Functions Solutions #1) quadratic #2) exponential

  48. Comparing Linear, Exponential, and Quadratic Functions Determine whether the table represents a linear, exponential, or quadratic function. • #5)

  49. Comparing Linear, Exponential, and Quadratic Functions Solutions • #5) linear

  50. Comparing Linear, Exponential, and Quadratic Functions • Tell whether the data represents a linear, exponential, or quadratic function. Then write the function. • #8) (-2, -9), (-1, 0), (0, 3), (1, 0), (2, -9)

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