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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) = ax 2 + bx + c OR ax 2 + bx + c = f(x)

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• 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)

Ex 2: How to Graph a Quadratic Function?

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

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)

Ex 2: How to Graph a Quadratic Function?

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

X Y

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

Ex 3: How to Graph a Quadratic Function?

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

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

Ex 4: How to Graph a Quadratic Function?

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

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

Ex 4: How to Graph a Quadratic Function?

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

Vertex goes here

f(x) = x2 – 8x + 17

a= , b= , c=

vertex: ( 4 , 1 )

2. Solving/

Finding Roots/

Finding Zeroes

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

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

Functions

(2n + 5)(n – 4) = 0

2. Solve for variable.

You will have two solutions/roots/zeroes

(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

1. Factor the polynomial and set the function = 0

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

f(x) = x2 – 4x + 3

2. Solving/

Finding Roots/

Finding Zeroes

Functions

1. Factor the polynomial and set the function = 0

x2 – 4x + 3 = 0

(x – 1)(x – 3) = 0

2. Solve for variable.

You will have two solutions/roots/zeroes

(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

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

x2 – 4x + 3 = 15

2. Solving/

Finding Roots/

Finding Zeroes

Functions

1. Factor the polynomial and set the function = 0

x2 – 4x + 3 = 15

-15 -15

x2 – 4x – 12 = 0

(x – 6)(x + 2) = 0

2. Solve for variable.

You will have two solutions/roots/zeroes

(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

Functions

1. Factor the polynomial and set the function = 0

x2 – 11x + 15 = -9

+9 +9

x2 – 11x + 24 = 0

(x – 8)(x – 3) = 0

2. Solve for variable.

You will have two solutions/roots/zeroes

(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

3. Find the Max/Min of a Quadratic Function

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( ) = ___

3. Find the Max/Min of a Quadratic Function

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( ) = ___

3. Find the Max/Min of a Quadratic Function

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: ( __ , __ )