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The Graph of aQuadratic

FunctionQuadratic Functions

Mr. Rolando B. Magat,Jr.

MAT - Math

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The graph of quadratic function f(x) = ax 2 + bx + c isthe same as the graph of the quadratic equationy = ax 2 + bx + c .

Properties of the Graph of f(x) = ax 2 + bx + c

1. If a is positive, then the parabola opens upward .

2. If a is negative, then the parabola opensdownward .

Graph of a Quadratic Function

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b

2a2

4ac b4a

3. The vertex has an x – coordinate of and

y – coordinate of .

Properties of the Graphof f(x) = ax

2

+ bx + c

4. The axis of symmetry is the vertical line . b

x2a

Graph of a Quadratic Function

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6. If x = h, then it is called the axis of symmetryand if y = k, then it is called the highest orlowest value of the function.

bh

2a

24ac bk

4a

5. The highest or lowest point of a parabola is thevertex of the parabola denoted by an

ordered pair (h, k) where

Properties of the Graphof f(x) = ax

2

+ bx + c

Graph of a Quadratic Function

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Graph the function f(x) = x 2 + 4x – 5. Determinethe vertex, the axis of symmetry, thehighest/lowest value, and the direction of theopening of the parabola.

Example:

Graph of a Quadratic Function

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Solution:

Step 1: Determine the coordinates of the vertex.

f(x) = x2 + 4x – 5

a = 1; b = 4; c = -5

2b 4ac bVertex ,

2a 4a

24 4(1)( 5) (4),

2(1) 4(1)

= (-2, -9)

Graph of a Quadratic Function

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Step 2: Construct the table of values.

-3

-4

-1

0

-5

-8

This is the vertex of theparabola. (-2, -9)-2 -9

-8

-5

(-4, -5)

(-3, -8)

(-1, -8)

(0, -5)

= (-4)2 + 4(-4) – 5

= (-3)2 + 4(-3) – 5

= (-1)2 + 4(-1) – 5

= (0)2 + 4(0) – 5

Graph of a Quadratic Function

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Step 3: Plot and connect the given pointsin the Cartesian plane.

Direction: Upward

Vertex: (-2, -9)

Axis of Symmetry: x = -2

Highest/Lowest Value: -9

Graph of a Quadratic Function

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Problem Set:

A. Determine the direction of the opening of thegraph, the vertex, axis of symmetry and thehighest/lowest value for each of the following

quadratic functions.

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B. Sketch the graph of the quadratic function f(x) =x2 – 8x + 16 and determine the direction of theopening of the graph, the vertex, axis of symmetryand the maximum/minimum value.

Graph of a Quadratic Function