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A Lesson on the Behavior of the Graphs of Quadratic Functions in the form y = a(x – h) 2 + k

A Lesson on the Behavior of the Graphs of Quadratic Functions in the form y = a(x – h) 2 + k. Objectives. The students should be able to explore the graphs of quadratic functions in the form y = a(x – h) 2 + k

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A Lesson on the Behavior of the Graphs of Quadratic Functions in the form y = a(x – h) 2 + k

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  1. A Lesson on the Behavior of the Graphs of Quadratic Functions in the form y = a(x – h)2 + k

  2. Objectives • The students should be able to explore the graphs of quadratic functions in the form y = a(x – h)2 + k • Analyze the effects of changes of a, h and k in the graphs of y = a(x – h)2 + k • Create a design using graphs of Quadratic Functions.

  3. Rewriting Quadratic Functions into Standard Form y = ax2 + bx + c y = x2 - 6x + 7 y = a(x – h)2 + k y = (x2 - 6x) + 7 y = (x2 - 6x + ) + 7 9 - 9 y = (x – 3)2 - 2

  4. y = a(x – h)2 + k y = x2 + 10x + 11 y = (x2 + 10x) + 11 y = (x2 + 10x + ) + 11 25 - 25 y = (x + 5)2 - 14

  5. y = a(x – h)2 + k y = 2x2 + 8x - 3 y = (2x2 + 8x) - 3 y = 2(x2 + 4x) - 3 y = 2(x2 + 4x + ) – 3 4 - 8 y = 2(x + 2)2 - 11

  6. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Lesson Proper or opens downward The graph of a quadratic function is a curve that either opens upward which are called parabola

  7. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 is called its vertex. • The point where the parabola changes its direction

  8. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Examine the behavior of the graphs as we change the sign of ain the function y = a(x – h)2 + k y = ¼ x 2 y = ½ x 2 - 4 y = -x 2 y = -2(x +5) 2

  9. Observation 1 Notice that if a is positive the parabola opensupward, otherwise it opens downward

  10. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Observe the behavior of the graphs as we change the value of ain the function y = a(x – h)2 + k y = 4 x 2 y = x 2 y = ½ x 2 y = ¼ x 2

  11. Observation 2 Notice that as we decrease the value of a, the opening of the parabola becomes wider

  12. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Study the behavior of the graphs as we change the value of h in the function y = a(x – h)2 + k y = ¼ (x – 0) 2 = ¼ x 2 y = ¼ (x – 8) 2 y = ¼ (x + 5) 2

  13. Observation 3 Notice that if h is positive the parabola is translated h units to the right whereas if h is negative the parabola is translated h units to the left

  14. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Monitor the behavior of the graphs as we change the value of k in the function y = a(x – h)2 + k y = -(x - 4) 2 + 0 = -(x – 4) 2 y = -(x - 4) 2 + 5 y = -(x - 4) 2 - 2

  15. Observation 4 Notice that if k is positive the parabola is translated k units upward whereas if k is negative the parabola is translated k units downward

  16. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Now, let us make a generalization on the behavior of the graph of y = ax 2 in relation to the graph ofy = a(x – h)2 + k Observe the graph of y = 4x 2in relation to the graph of y = 4(x + 9)2 + 4 and y = 4(x – 4)2 - 6

  17. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Another Example Discuss relationships between the graphs ofy = - 2(x + 4)2 -3 and y = - 2(x + 1)2 + 7 in relation to the graph of of y = 2x 2

  18. y 6 4 2 -2 -4 -6 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 More Example y = x2 Graph y = (x - 8)2 - 5 and compare its characteristicsto y = x2 y = (x - 8)2 - 5

  19. Summary • side upward/ downward translation value of a (h, k) sideward translation sign of a • size of the opening • value of h • value of k • vertex

  20. Evaluation • Which of the following has a narrower opening of its parabola? • y = 3x2 – 2 • y = ½ (x – 5)2 – 2 • y = - ¼ (x + 3)2 + 8 2. Which of the following opens upward? a. y = -3x2 – 2 b. y = ½ (x – 5)2 – 2 c. y = - ¼ (x + 3)2 + 8

  21. 3. With respect to the graph of y = 4x2 , the graph of y = 4(x – 5)2 + 6 is translated 5 units to the (left, right) 4. With respect to the graph of y = - ½ x2 , the graph of y = - ½ (x + 3)2 - 9 is translated how many units downward? 5. At which quadrant can we locate the vertex of y = (x – 1)2 + 6?

  22. Student’s Output Draw a picture using graphs of Quadratic Functions • and make a discussion guided with the following questions; • What is the design? Give a title to your design. • What are the characteristics of the graph of Quadratic Functions that you considered in order to complete the design? ( translation, increasing the value of a, changing the sign of a, …) • Share some insights of your new learnings in making the project.

  23. Assignment • How do you determine the zeros of the function? • What is meant by the zeros of the function? • Find the zeros of 1. y = x2 - 6x + 7 2. y = 2(x + 2)2 - 11

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