MATHEMATICS: QUADRATIC FUNCTION & GRAPHS OF QUADRATIC FUNCTION, Slides of Mathematics

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QUADRATIC

FUNCTION

DEFINITION

The function f(x) = +bx + c is called

a quadratic function.

Let a, b, c be real numbers with a 0.

The graph of a quadratic function is a special type of “U”- shaped curve that is called a parabola. ( U shaped) All parabolas are symmetric with respect to a line called the axis of symmetry , or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola. Quadratic Function

f (x)= y = ax ² + bx + c Standard form Quadratic Equations y = a(x-h)² + k Vertex form

The graph is “U=shaped”

and is called a parabola.

The axis of

symmetry for the

parabola is the vertical

line through he vertex.

y=x²

Opens up

y=-x²

Opens down

Example:

Graph f(x)= x² - 2

x f(x)= x²-

Example:

Comparison of functional values

f(x)= x² and f(x)= 2x²

x

f(x)= x² f(x)= 2x²

• 0 1 2 f(-2)= (-2)² = 4

f(x)= x²

f(-1)= (-1)² = 1 f(0)= (0)² = 0 f(1)= (1)² = 1 f(2)= (2)² = 4

f(x)=

f(-2)= 2x² 2(-2)²

f(-1)= 2(-1)² = 2 f(0)= 2(0)² = 0 f(1)= 2(1)² = 2 f(2)= 2(2)² = 8

x f(x)=

x² f(x)= 2x² -2 4 8 -1 1 2 0 0 0 1 1 2 2 4 8

Example:

Graph f(x)= -x²

x f(x)= -x²

f(0)= -(0)²

f(1)= -(1)²

f(2)= -(2)²

f(-1)= -(-1)²

f(-2)= -(-2)²

x f(x)= -x²

Example:

Graph f(x)= -x²

x f(x)= -x²

Example:

Graph f(x)= (x-3)²

x f(x)= (x-

SUMMARY OF GRAPHING FUNCTION

f(x)= x²

f(x)= x² + k

f(x)= a x²

f(x)= ( x – h )²

basic parabola moves parabola up and down affects the width and the way parabola opens moves parabola right or left

Example:

Graph f(x)= x² - 4x + 3

Solution:

f(x)= x² - 4x + 3

= (x² - 4x) + 3

= (x² - 4x + 4 ) + 3

Add 4, which is the square of one-half of the coefficient of x.

= (x² - 4x + 4 ) + 3 -

Subtract 4 to compensate for the 4 that was added.

= (x - 2)² - 1

Solution:

f(x)= (x - 2)² - 1

moves the parabola 2 units to the right moves the parabola 1 unit down

The graph of f(x)= (x-2)² - 1 is the

basic parabola moved two units to

the right and one unit down.