# A Lesson on the Behavior of 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

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

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

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

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

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

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

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

Notice that if a is positive the parabola opens upward, otherwise it opens downward Observation 1

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

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

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

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 Observation 3

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

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 Observation 4

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

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

More Example -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 6 4 2 -2 -4 -6 x y y = (x - 8) 2 - 5 and compare its characteristics to y = x 2 Graph y = (x - 8) 2 - 5 and compare its characteristics to y = x 2 y = x 2 y = (x - 8) 2 - 5

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

Evaluation 1. Which of the following has a narrower opening of its parabola? a. y = 3x 2 – 2 b. y = ½ (x – 5) 2 – 2 c. 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

3. With respect to the graph of y = 4x 2, 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 x 2, 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?

Student’s Output Draw a picture using graphs of Quadratic Functions and make a discussion guided with the following questions; and make a discussion guided with the following questions; What is the design? Give a title to your design.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, …)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.Share some insights of your new learnings in making the project.

Title(20 pts) The title is original and impressive (18 pts) The title is original (16 pts) The title is not original but still gives an impact (14 pts) The title is irrelevant Creativity(50 pts) The design shows creativity, cleanliness and the choice of colors enhance the presentation of the design (45 pts) The design shows creativity and the choice of colors enhance the presentation of the design (40 pts) The design shows less creativity and the choice of colors does not help enhance the presentation. (35 pts) The design shows less effort. Obviously it was done for the purpose of submission only. Content/ Discussion (30 pts) 100% of the content is correct (27 pts) There are at most 2 statements which are not correct (24 pts) One half of the content are not correct (21 pts) Majority of the content are not correct

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

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