Graphs of Quadratic Function

• Introducing the concept: Transformation of the Graph of y = x2

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f x = x2

Graph of f(x) = ax2 and a(x-h)2

• Objective: Graph a function f(x)=a(x-h)2, and determine its characteristics.

Definition: A QUADRATIC FUNCTION is a function that can be described as f(x) = ax2 + bc + c 0.

Graphs of QUADRATIC FUNCTIONS are called PARABOLAS.

Now let us see the graphs of quadratic functions

Graph of QUADRATIC FUNCTION

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-2

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h x = 0.5x2

g x = 2x2

f x = x2

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-2

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h x = -0.5x2

g x = -2x2

f x = -x2

LINE , OR AXIS OF SYMMETRY

VERTEX

LINE , OR AXIS OF SYMMETRY

VERTEX

• Thus the y-axis is the LINE SYMMETRY. The point (0,0) where the graph crosses the line of symmetry, is called VERTEX OF THE PARABOLA

Next consider f(x) = ax2, we know the following about its graph. Compared with the graph of f(x) = x2.

1. If > 1, the graph is stretched vertically.

2. If < 1, the graph is shrunk vertically.

3. If a < 0, the graph is reflected across the x-axis.

a

aa

EXAMPLE:a. Graph f(x) =3x2

b. Line of Symmetry? Vertex?

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f x = 3x2

LINE OF SYMMETRY

The y-axisVERTEX

(0,0)

Exercise:a. Graph f(x) = -1/4 x2

b. Line of symmetry and Vertex?• Your answer should be like this

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f x = -x2

4

LINE OF SYMMETRY

Y-AXIS

VERTEX

(0,0)

In f(x) = ax2, let us replace x by x – h. if h is positive, the graph will be translated to the right. If h is negative the translation

will be to the left. The line, or axis of symmetry and the vertex will also be

translated the same way. Thus f(x) = a(x-h)2, the axis of symmetry is x = h and

the vertex is (h, 0).

Compare the Graph of f(x) = 2(x+3)2 to the graph of f(x) = 2x2.

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f x = 2x+3 2

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f x = 2x2

VERTEX (0,3)

LINE OF SYMMETRY, X = -3 VERTEX (0,0),

SYMMETRY, Y-AXIS

EXAMPLE:a. Graph f(x) = - 2(x-1)2

b. Line of Symmetry and Vertex?2

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f x = -2x-1 2VERTEX (h, 0) = (1,0)

LINE OF SYMMETRY, X=1

EXERCISES:a. Graph f(x) = 3(x-2)2

b. Line of Symmetry and Vertex?6

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f x = 3x-2 2

LINE OF SYMMETRY, X=2

VERTEX (2,0)

Graph of f(x) = a(x-h)2+k• Objective: Graph a function f(x) = a(x-h)2 +

k, and determine its characteristics.

In f(x) = a(x-h)2, let us replace f(x) by f(x) – k

f(x) – k = a(x-h)2

Adding k on both sides gives f(x) = a(x-h)2 + k.

The Graph will be translated UPWARD if k is Positive and DOWNWARD if k is NEGATIVE. The Vertex will be translated the same way. The Line of Symmetry will NOT be AFFECTED

Guidelines for Graphing Quadratic Functions,

f(x)=a(x-h)2 + k• When graphing quadratic function in the form

f(x)=a(x-h)2+k,1. The line of symmetry is x-h=0, or x = h.2. The vertex is (h,k).3. If a > 0, then (h,k) is the lowest point of the graph, and

k is the MINIMUM VALUE of the function.4. If a < 0, then (h,k) is the highest point of the graph,

and k is the MAXIMUM VALUE of the function.

Example:a. Graph f(x) = 2(x+3)2 – 2b. Line of Symmetry, Vertex?c. is there a min/max value? If so, what is it?

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f x = 2x+3 2-2

LINE OF SYMMETRY, X=-3

VERTEX: ( -3,-2)

MINIMUM: -2

Exercises:for each of the following, graph the function, find the vertex, find the line of symmetry, and find the min/ max value.

• 1. f(x) = 3(x-2)2 + 4

• 2. f(x) = -3(x+2)2 - 4

Answer #1

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f x = 3x-2 2+4

VERTEX: (2,4)

MIN: 4

LINE OF SYMMETRY:X =2

Answer #24

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f x = -3x+2 2-1

VERTEX: (-2,-1)

MAX: -1

LINE OF SYMMETRY:X = -2

ANALYZING f(x) = a(x-h)2+k

• Objective: Determine the characteristics of a function f(x) = a(x-h)2+k

EXAMPLE:Without graphing, find the vertex,line of symmetry, min/max value.Given:1. f(x) = 3(x-1/4)2+42. g(x) = -4x+5)2+7

a. What is the Vertex?

b. Line of Symmetry?

c. Is there a Min / Max Value?

d. What is the min / max value?

Answer in #1 and #2

a. What is the Vertex?

#1. (1/4, -2) #2. ( -5, 7)

b. Line of Symmetry?

X = ¼ X = -5

c. Is there a Min / Max Value?

Minimum. The graph extends upward since 3>0

Maximum. The graph extends downward since –4<0.

d. What is the min / max value?

Min.Value is –2 Max.Value is 7