Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions The quadratic function...

Evaluating and Graphing Quadratic Functions

1. Graphing Quadratic Functions

• The quadratic function is a second-order polynomial function

• It is always written in this format, with the coefficient parameters: a, b, c

f(x) = ax2 + bx + c

OR

ax2 + bx + c = f(x)



• Example 1: Identify the coefficient parameters (a, b, c) in the following quadratic functions:a. f(x) = 5x2 + 10x + 15

b. f(n) = n2 – 10n + 22

c. h(r) = 7r2 + 14r – 7

d. g(x) = x2 – 25

e. g(x) = 7x2

f. h(n) = 5x – 24

a = 5, b = 10, c = 15

a = 1, b = -10, c = 22

a = 7, b = 14, c = -7

a = 1, b = 0, c = -25

a = 7, b = 0, c = 0

a = 0, b = 5, c = -24 (linear)



• This is a graph of a quadratic function. We call it a parabola

• What are some observations we can make about the graph?

x

y



• The axis of symmetry is the x-coordinate that forms the middle and splits the parabola in two halves• Axis of Symmetry: x = - b

Formula 2a

• The vertex is the (x,y) ordered pair on the axis of symmetry• Plug in the axis of symmetry for x to

find the y coordinate



1. Find axis of symmetry• X = - b

2a

2. Find the vertex• plug in the axis of

symmetry for x to find the y-coordinate of the vertex

Ex 2: How to Graph a Quadratic Function?

1. f(x) = x2 – 10x + 24a=1, b=-10, c=24

x = -b = -(-10) = 10 = 5 2a 2(1) 2

2. f(x) = x2 – 10x + 24 f(5) = (5)2 – 10(5) + 24f(5) = 25 – 50 + 24f(5) = -1

vertex: (5, -1)

1. Identify a, b, c

2. Find the axis of symmetry

3. Find the vertex (x,y)

4. Fill in data table values

5. Graph the function



3. Make a data table (x/y or in/out table)

• Put the vertex in the middle value

• Plug-in two x-values lower and two x-values higher than the axis of symmetry

• You should notice some symmetry in your output values

X Y

3

4

5 -1

6

7

f(x) = x2 – 10x + 24a=1, b=-10, c=24vertex: (5, -1)


1. Identify a, b, c







4. First plot the vertex. Then plot all the other points on your graph. Draw your parabola. Done!

X Y

3 3

4 0

5 -1

6 0

7 3

f(x) = x2 – 10x + 24a=1, b=-10, c=24vertex: (5, -1)


1. Identify a, b, c







X Y

f(x) = x2 – 6x + 8a= , b= , c= vertex: ( __ , __ )


1. Identify a, b, c





x = -b = = ___ 2a

f(x) = x2 – 6x + 8f( ) = ( )2 – 6( ) + 8

f( ) = ___

Vertex goes here



f(x) = (x – 4)2 + 1

f(x) = (x – 4)(x – 4) + 1

f(x) = x2 – 8x + 16 + 1

f(x) = x2 – 8x + 17


1. Identify a, b, c





Use the box method to multiply the binomials

x2 -4x

-4x 16

x -4

x

-4

f(x) = x2 – 8x + 17

a= , b= , c=

vertex: ( __ , __ )

x = -b = = ___ 2a

f(x) = x2 – 8x + 17f( ) = ( )2 – 8( ) + 17

f( ) = ___



X Y


1. Identify a, b, c





Vertex goes here

f(x) = x2 – 8x + 17

a= , b= , c=

vertex: ( 4 , 1 )


2. Solving/

Finding Roots/

Finding Zeroes

Of Quadratic

Functions

What is true about where the curve intercepts the x-axis? How many times does it intercept the x-axis?

x

y

• Y = f(x) = 0 at the x-intercepts (curve crosses x-axis)

• The x-coordinates where y=0 are called solutions, or the roots, or the zeroes of the quadratic function


• I DO: Find the solution of the following function:

f(x) = (x – 3)(x + 2)

2. Solving/

Finding Roots/

Finding Zeroes

Of Quadratic

Functions

(x – 3)(x + 2) = 0

2. Solve for variable.

You will have two solutions/roots/zeroes

(in most cases)

(x – 3)(x + 2) = 0

x – 3 = 0

+3 +3

x = 3

x + 2 = 0

-2 -2

x = -2

3. Write the solutionSolutions: (3,0) and (-2,0)

The parabola crosses the x-axis at x = -2 and x = 3

1. Factor the polynomial and set the function = 0


• WE DO: Find the solution of the following function:

f(n) = (2n + 5)(n – 4)

2. Solving/

Finding Roots/

Finding Zeroes

Of Quadratic

Functions

(2n + 5)(n – 4) = 0



(in most cases)

(2n + 5)(n – 4) = 0

2n + 5 = 0

-5 -5

2n = -5

2 2

n = -2.5

n – 4 = 0

+4 +4

x = 4

3. Write the solutionSolutions: (-2.5, 0) and (4, 0)

The parabola crosses the x-axis at x = -2.5 and x = 4




f(x) = x2 – 4x + 3

2. Solving/

Finding Roots/

Finding Zeroes

Of Quadratic

Functions


x2 – 4x + 3 = 0

(x – 1)(x – 3) = 0



(in most cases)

(x – 1)(x – 3) = 0

x – 1 = 0

+1 +1

x = 1

x – 3 = 0

+3 +3

x = 3

3. Write the solutionSolutions: (1,0) and (3,0)

The parabola crosses the x-axis at x = 1 and x = 3



x2 – 4x + 3 = 15

2. Solving/

Finding Roots/

Finding Zeroes

Of Quadratic

Functions


x2 – 4x + 3 = 15

-15 -15

x2 – 4x – 12 = 0

(x – 6)(x + 2) = 0



(in most cases)

(x – 6)(x + 2) = 0

3. Write the solutionSolutions: (-2,0) and (6,0)

The parabola crosses the x-axis at x = -2 and x = 6

x – 6 = 0

+6 +6

x = 6

x + 2 = 0

-2 -2

x = -2


• WE DO: Find the solution of the following function:

x2 – 11x + 15 = -9

2. Solving/

Finding Roots/

Finding Zeroes

Of Quadratic

Functions


x2 – 11x + 15 = -9

+9 +9

x2 – 11x + 24 = 0

(x – 8)(x – 3) = 0



(in most cases)

(x – 8)(x – 3) = 0

3. Write the solutionSolutions: (3,0) and (8,0)

The parabola crosses the x-axis at x = 3 and x = 8

x – 8 = 0

+8 +8

x = 8

x – 3 = 0

+3 +3

x = 3


3. Find the Max/Min of a Quadratic Function

What is Concavity?

f(x) = x2 – 4x – 5a= 1, b=-4, c=-5

f(x) = x2

a= 1, b=0, c=0

a= -1, b=0, c=0

f(x) = -x2 – 2x + 3a= -1, b=-2, c=3



What is Concavity?

Coefficient Parameter “a”

Concavity Shape of Parabola

Vertex is Max or Min?

a > 0 Concave UpVertex is Minimum

a < 0Concave Down

Vertex is Maximum

A = 0Linear

No Concavity

No max/min

No vertex


f(x) = x2 – 6x + 8a= , b= , c=

Ex 1 (I DO): What is the vertex of the quadratic function? Is it a relative max or min?

x = -b = = ___ 2a

f(x) = x2 – 6x + 8f( ) = ( )2 – 6( ) + 8

f( ) = ___


What is Concavity?

Is it a max or min? Is a>0 or a<0?

a=1, greater than 0, concave up, vertex is minimum

The minimum of -1 is where x = 3

vertex: ( __ , __ )


f(x) = -x2 + 8x – 20a= , b= , c=

Ex 2 (WE DO): What is the vertex of the quadratic function? Is it a relative max or min?

x = -b = = ___ 2a

f(x) = -x2 + 8x – 20 f( ) = -( )2 + 8( ) – 20

f( ) = ___


What is Concavity?

Is it a max or min? Is a>0 or a<0?

a=-1, less than 0, concave down, vertex is maximum

The maximum of -4 is where x = 4

vertex: ( __ , __ )

Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions The quadratic function...

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