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

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.

Rewriting Quadratic Functions into Standard

Formy = ax2 + bx + c

y = a(x – h)2

+ k

y = x2 - 6x + 7y = (x2 - 6x) + 7y = (x2 - 6x + ) + 7 y = (x – 3)2 - 2

9 - 9

y = x2 + 10x + 11y = (x2 + 10x) + 11y = (x2 + 10x + ) + 11 y = (x + 5)2 - 14

25

- 25

y = a(x – h)2

+ k

y = 2x2 + 8x - 3

y = (2x2 + 8x) - 3

y = 2(x2 + 4x + ) – 3 y = 2(x + 2)2 - 11

y = 2(x2 + 4x) - 3

4 - 8

y = a(x – h)2

+ k

The graph of a quadratic function is a curve that either opens upwardor 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

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

6

4

2

-2

-4

-6

x

yis 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 2y = 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) 2y = 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

+ 5y = -(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

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

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

yGraph y = (x - 8)2 - 5 and compare its characteristics to y = x2

y = x2

y = (x - 8)2 - 5

Summary

side

size of the opening

value of h

value of k

vertex

upward/ downward translation

value of a

(h, k)

sideward translation

sign of a

Evaluation1. Which of the following has a

narrower opening of its parabola?a. y = 3x2 – 2b. y = ½ (x – 5)2 – 2c. y = - ¼ (x + 3)2 + 8

2. Which of the following opens upward?

a. y = -3x2 – 2b. y = ½ (x – 5)2 – 2c. y = - ¼ (x + 3)2 + 8

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?

Student’s OutputDraw 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.

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 presentatio

n of the design

(45 pts)The design

shows creativity and the

choice of colors

enhance the presentatio

n of the design

(40 pts)The design

shows less creativity and the

choice of colors does

not help enhance the

presentatio

n.

(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?

What is meant by the zeros of the function?

Find the zeros of 1. y = x2 - 6x + 72. y = 2(x + 2)2 - 11