1. 1. The quadratic function is a second-order polynomial function
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)

2. 1.
Graphing Quadratic Functions
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)

3. 1. This is a graph of a quadratic function. We call it a parabola
Graphing Quadratic Functions
This is a graph of a quadratic function. We call it a parabola
What are some observations we can make about the graph? y x

4. 1.
Graphing Quadratic Functions
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

5. 1. Ex 2: How to Graph a Quadratic Function? f(x) = x2 – 10x + 24
Graphing Quadratic Functions
Ex 2: How to Graph a Quadratic Function?
f(x) = x2 – 10x + 24
a=1, b=-10, c=24
x = -b = -(-10) = 10 = 5
2a 2(1)
2. f(x) = x2 – 10x + 24
f(5) = (5)2 – 10(5) + 24
f(5) = 25 –
f(5) = -1
vertex: (5, -1)
Find axis of symmetry
X = - b 2a
Find the vertex
plug in the axis of symmetry for x to find the y-coordinate of the vertex
Identify a, b, c
Find the axis of symmetry
Find the vertex (x,y)
Fill in data table values
Graph the function

6. 1. Ex 2: How to Graph a Quadratic Function? X Y 3 4 5 -1 6 7
Graphing Quadratic Functions
Ex 2: How to Graph a Quadratic Function?
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
f(x) = x2 – 10x + 24
a=1, b=-10, c=24
vertex: (5, -1) X Y 3 4 5 -1 6 7
Identify a, b, c
Find the axis of symmetry
Find the vertex (x,y)
Fill in data table values
Graph the function

7. 1. Ex 2: How to Graph a Quadratic Function? X Y 3 4 5 -1 6 7
Graphing Quadratic Functions
Ex 2: How to Graph a Quadratic Function?
4. First plot the vertex. Then plot all the other points on your graph. Draw your parabola. Done!
f(x) = x2 – 10x + 24
a=1, b=-10, c=24
vertex: (5, -1)
Identify a, b, c
Find the axis of symmetry
Find the vertex (x,y)
Fill in data table values
Graph the function X Y 3 4 5 -1 6 7

8. 1. Ex 3: How to Graph a Quadratic Function? X Y f(x) = x2 – 6x + 8
Graphing Quadratic Functions
Ex 3: How to Graph a Quadratic Function?
f(x) = x2 – 6x + 8
a= , b= , c=
vertex: ( __ , __ )
f(x) = x2 – 6x + 8
f( ) = ( )2 – 6( ) + 8
f( ) = ___
x = -b = = ___ 2a
Identify a, b, c
Find the axis of symmetry
Find the vertex (x,y)
Fill in data table values
Graph the function X Y
Vertex goes here

9. 1. Ex 4: How to Graph a Quadratic Function? x2 -4x 16
Graphing Quadratic Functions
Ex 4: How to Graph a Quadratic Function?
f(x) = (x – 4)2 + 1
f(x) = (x – 4)(x – 4) + 1
f(x) = x2 – 8x
f(x) = x2 – 8x + 17
Use the box method to multiply the binomials x -4 x x2
-4x 16
Identify a, b, c
Find the axis of symmetry
Find the vertex (x,y)
Fill in data table values
Graph the function -4
f(x) = x2 – 8x + 17
a= , b= , c=
vertex: ( __ , __ )
f(x) = x2 – 8x + 17
f( ) = ( )2 – 8( ) + 17
f( ) = ___
x = -b = = ___ 2a

10. 1. Ex 4: How to Graph a Quadratic Function? X Y f(x) = x2 – 8x + 17
Graphing Quadratic Functions
Ex 4: How to Graph a Quadratic Function?
f(x) = x2 – 8x + 17
a= , b= , c=
vertex: ( 4 , 1 )
Identify a, b, c
Find the axis of symmetry
Find the vertex (x,y)
Fill in data table values
Graph the function X Y
Vertex goes here

11. 2.
Solving/
Finding X-Intercepts
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?
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 y x

12. The parabola crosses the x-axis at x = -2 and x = 32.
Solving/
Finding X-Intercepts
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
I DO: Find the solution of the following function:
f(x) = (x – 3)(x + 2)
1. Factor the polynomial and set the function = 0
(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
x = 3
x + 2 = 0
x = -2
Solutions: (3,0) and (-2,0)
The parabola crosses the x-axis at x = -2 and x = 3
3. Write the solution

13. The parabola crosses the x-axis at x = -2.5 and x = 4
Solving/
Finding X-Intercepts
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
WE DO: Find the solution of the following function:
f(n) = (2n + 5)(n – 4)
1. Factor the polynomial and set the function = 0
(2n + 5)(n – 4) = 0
(2n + 5)(n – 4) = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
2n + 5 = 0
2n = -5
n = -2.5
n – 4 = 0
x = 4
Solutions: (-2.5, 0) and (4, 0)
The parabola crosses the x-axis at x = -2.5 and x = 4
3. Write the solution

14. The parabola crosses the x-axis at x = 1 and x = 32.
Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
I DO: Find the solution of the following function:
f(x) = x2 – 4x + 3
x2 – 4x + 3 = 0
(x – 1)(x – 3) = 0
1. Factor the polynomial and set the function = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 1)(x – 3) = 0
x – 1 = 0
x = 1
x – 3 = 0
x = 3
Solutions: (1,0) and (3,0)
The parabola crosses the x-axis at x = 1 and x = 3
3. Write the solution

15. The parabola crosses the x-axis at x = -2 and x = 62.
Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
I DO: Find the solution of the following function:
x2 – 4x + 3 = 15
x2 – 4x + 3 = 15
x2 – 4x – 12 = 0
(x – 6)(x + 2) = 0
1. Factor the polynomial and set the function = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 6)(x + 2) = 0
x – 6 = 0
x = 6
x + 2 = 0
x = -2
Solutions: (-2,0) and (6,0)
The parabola crosses the x-axis at x = -2 and x = 6
3. Write the solution

16. The parabola crosses the x-axis at x = 3 and x = 82.
Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
WE DO: Find the solution of the following function:
x2 – 11x + 15 = -9
x2 – 11x + 15 = -9
x2 – 11x + 24 = 0
(x – 8)(x – 3) = 0
1. Factor the polynomial and set the function = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 8)(x – 3) = 0
x – 8 = 0
x = 8
x – 3 = 0
x = 3
Solutions: (3,0) and (8,0)
The parabola crosses the x-axis at x = 3 and x = 8
3. Write the solution

17. 3. Find the Max/Min of a Quadratic Function What is Concavity?
f(x) = x2 – 4x – 5
a= 1, b=-4, c=-5
f(x) = x2
a= 1, b=0, c=0
f(x) = -x2 – 2x + 3
a= -1, b=-2, c=3
a= -1, b=0, c=0

18. Coefficient Parameter “a”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 Up
Vertex is Minimum
a < 0
Concave Down
Vertex is Maximum
A = 0
Linear
No Concavity
No max/min
No vertex

19. 3.
Find the Max/Min of a Quadratic Function
What is Concavity?
Ex 1 (I DO): What is the vertex of the quadratic function? Is it a relative max or min?
f(x) = x2 – 6x + 8
a= , b= , c=
f(x) = x2 – 6x + 8
f( ) = ( )2 – 6( ) + 8
f( ) = ___
x = -b = = ___ 2a
vertex: ( __ , __ )
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

20. 3.
Find the Max/Min of a Quadratic Function
What is Concavity?
Ex 2 (WE DO): What is the vertex of the quadratic function? Is it a relative max or min?
f(x) = -x2 + 8x – 20
a= , b= , c=
f(x) = -x2 + 8x – 20
f( ) = -( )2 + 8( ) – 20
f( ) = ___
x = -b = = ___ 2a
vertex: ( __ , __ )
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