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5 – 1: Graphing Quadratic Functions (Day 1 )

In this lesson, students will learn to graph quadratic functions and determine the maxima, minima, and zeros. They will understand the characteristics of a parabola and how to identify the vertex and axis of symmetry. Three forms of quadratic functions will be covered: standard form, vertex form, and intercept form.

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5 – 1: Graphing Quadratic Functions (Day 1 )

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  1. 5 – 1: Graphing Quadratic Functions(Day 1 ) Objective: CA 10: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

  2. The graph of a quadratic function is U – shaped and is called a parabola. The graph of y = x2 and y = - x2 are shown.

  3. The origin is at the bottom of the graph y = x2 and the highest point of the graph y = - x2. The lowest or highest point on the graph of a quadratic function is called the vertex.

  4. The graphs of y = x2 and y = - x2are symmetric about the y – axis, called the axis of symmetry. In general, the axis of symmetry for the graph is the vertical line through the vertex.

  5. Quadratic functions have 3 forms: #1. A quadratic function has the form (standard form) where a ≠ 0.

  6. The graph of a quadratic function is a parabola with these characteristics: • The parabola opens up if a > 0 and opens down if a < 0. • The parabola is wider than the graph of y = x2 if |a| < 1 and narrower than the graph of y = x2 if |a| > 1

  7. (characteristics continued) The x-coordinate of the vertex is The axis of symmetry is the vertical line

  8. #2. • Characteristics of graph: • The vertex is (h, k) • The axis of symmetry is x = h

  9. #3. • Characteristics of graph: • The x – intercepts are p and q. • The axis of symmetry is halfway between (p, 0) and (q, 0).

  10. Example 1: Graphing a Quadratic Function. Graph 1. Coefficients for this function are: a = 2 b = -8 c = 6 2. Since a > 0 the parabola opens upward.

  11. 3. Find and plot the vertex. The y - coordinate is: Vertex = (2, -2) So the vertex is (2, -2) Draw the axis of symmetry x = 2

  12. 4. Draw the axis of symmetry x = 2

  13. Plot two points on one side of the axis of symmetry, such as (1, 0) and (0, 6). Use symmetry to plot two more points such as (3, 0) and (4, 6).

  14. Draw the parabola through the points.

  15. Example 2: Graphing a Quadratic function in Vertex from. Graph What we know: graph opens downward because a < 0.

  16. The vertex is (-3, 4) The A.o.S is x = - 3 (-3,4)

  17. Graphing a Quadratic function in Intercept form. Graph Intercept Form: From observation we know the following The parabola opens downward

  18. The x – intercepts occur at: (-2, 0) and (4, 0) The axis of symmetry lies half way between –2 and 4 which is x = 1

  19. Example 4: Write the quadratic function in standard form.

  20. Write the quadratic function in standard form.

  21. Homework: Page 253 #17 – 19, #21 – 43 odd

  22. Investigating Parabolas page 249 1. Use a graphing calculator to graph each of these functions in the same viewing windows: 2. Repeat Step 1 for these functions:

  23. 3. What are the vertex and the axis of symmetry of the graph of y = ax2? (0, 0); x = 0 • Describe the effect of a on the graph of y = ax2? The graph opens up if a > 0, the graph opens down if a < 0.

  24. By observation we know the following about this function. this means that the graph opens downward because a < 0. The vertex is (-3, 4). The axis of symmetry is x = - 3

  25. To graph the function: plot the vertex (-3, 4). Draw the axis of symmetry x = -3

  26. Plot two points to the right such as (-1, 2) and (1, -4). Use the axis of symmetry to plot two points to the left (-5, 2) and (-7, -4 )

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