1. 5.1 Modeling Data with Quadratic Functions
Quadratic Functions and Their Graphs

2. 1) Quadratic Formulas and Their Graphs
A quadratic function is a function that produces a parabola.

3. 1) Quadratic Formulas and Their Graphs
A quadratic function is a function that produces a parabola.

4. 1) Quadratic Formulas and Their Graphs
A quadratic function is a function that produces a parabola.

5. 1) Quadratic Formulas and Their Graphs
The equation of a quadratic function can be written in standard form. 
Quadratic term
Linear term
Constant term

6. Quadratic Function: f(x) = ax2 + bx + c ‘a’ cannot = 0

7. 1) Quadratic Formulas and Their Graphs
Since the largest exponent of function is 2, we say that a quadratic equation has a degree of 2. 
Equations of second degree are called quadratic.

8. QUADRATICS - - what are they?
Y = ax² bx c
FORM _______________________
Quadratic
term
Linear
term
Constant
Important Details
c is y-intercept
a determines shape and position
if a > 0, then opens up
if a < 0, then opens down
Vertex: x-coordinate is at –b/2a

9. Parts of a parabola These are the roots Roots are also called: -zeros
-solutions
- x-intercepts
This is the y-intercept, c
It is where the parabola
crosses the y-axis
This is the vertex, V
This is the called
the axis of symmetry, a.o.s.
Here a.o.s. is the line x = 2

10. STEPS FOR GRAPHING Y = ax² + bx +c
HAPPY or SAD ?
2 VERTEX = ( -b / 2a , f(-b / 2a) )
T- Chart
Axis of Symmetry

11. GRAPHING - - Standard Form (y = ax² + bx + c) y = x² + 6x + 8
The graph will be symmetrical. Once you have half the graph, the other two points come from the mirror of the first set of points.
1) It is happy because a>0
FIND VERTEX (-b/2a)
a =1 b=6 c=8
So x = -6 / 2(1) = -3
Then y = (-3)² + 6(-3) + 8 = -1
So V = (-3 , -1)
3) T-CHART
X Y = x² + 6x + 8
Why -2 and 0?
Pick x values where the graph will cross an axis -2
y = (-2)² + 6(-2) + 8 = 0
y = (0)² + 6(0) + 8 = 8 11

12. GRAPHING - - Standard Form (y = ax² + bx + c) y = -x² + 4x - 5
1) It is sad because a<0
FIND VERTEX (-b/2a)
a = b=4 c=-5
So x = - 4 / 2(-1) = 2
Then y = -(2)² + 4(2) – 5 = -1
So V = (2 , -1)
Here we only have one point where the graph will cross an axis. Choose one other point (preferably between the vertex and the intersection point) to graph.
3) T-CHART
X Y = -x² + 4x - 5 1
y = -(1)² + 4(1) - 5 = -2
y = -(0)² + 4(0) – 5 = -5 12

13. 1) Quadratic Formulas and Their Graphs
Example 1:
Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.

14. 1) Quadratic Formulas and Their Graphs
Example 1:
Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.
This IS a quadratic function.
QUADRATIC TERM: x2
LINEAR TERM: 3x
CONSTANT TERM: none

15. 1) Quadratic Formulas and Their Graphs
Example 1:
Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.
This IS a quadratic function.
QUADRATIC TERM: x2
LINEAR TERM: 3x
CONSTANT TERM: none
This is NOT a quadratic function.
QUADRATIC TERM: none
LINEAR TERM: 5x
CONSTANT TERM: none

16. Ex 1 Is the function linear or quadratic? f(x) = (2x – 1)2
The function is quadratic because “a” (the coefficient of x squared) is equal to 4. If “a” were equal to zero, then the function would be linear.

17. EX 2
Is the function linear or quadratic?
f(x) = x2 – (x + 1)(x – 1)

18. EX 2 Is the function linear or quadratic? f(x) = x2 – (x + 1)(x – 1)

19. EX 2 Is the function linear or quadratic? f(x) = x2 – (x + 1)(x – 1)

20. EX 2 Is the function linear or quadratic? f(x) = x2 – (x + 1)(x – 1)
The function is linear.

21. 1) Quadratic Formulas and Their Graphs
Example 3:
Identify the vertex and axis of symmetry for the parabola. Identify points corresponding to P and Q. 3 P 2 1 -2 -1 1 Q 2 3 4 -1 -2

22. 1) Quadratic Formulas and Their Graphs
Example 3:
Identify the vertex and axis of symmetry for each parabola. Identify points corresponding to P and Q. 3
Vertex: (1, -1)
Axis of symmetry: x = 1
P’ (3, 3)
Q’ (0, 0) P P P’ 2 1 -2 -1 1 Q Q’ 2 3 4 -1 -2

23. Ex 4
y = 2x2 + x – c contains the point (1, 2). Find c.

24. Ex 4 y = 2x2 + x – c contains the point (1, 2). Find c.

25. Ex 4 y = 2x2 + x – c contains the point (1, 2). Find c.

26. Ex 4 y = 2x2 + x – c contains the point (1, 2). Find c.

27. Ex 4 y = 2x2 + x – c contains the point (1, 2). Find c.

28. 1) Quadratic Formulas and Their Graphs
We can graph parabolas using a table of values.

29. 1) Quadratic Formulas and Their Graphs
We can graph parabolas using a table of values.
Recall…graphing linear functions…

30. 1) Quadratic Formulas and Their Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of values.

31. 1) Quadratic Formulas and Their Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of values. x y -2 -1 1 2

32. 1) Quadratic Formulas and Their Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of values. x y -2
(-2)2 = 4 -1
(-1)2 = 1
(0)2 = 0 1
(1)2 = 1 2
(2)2 = 4

33. 1) Quadratic Formulas and Their Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of values. x y -2 4 -1 1 2

34. 1) Quadratic Formulas and Their Graphs
The axis of symmetry is a line that divides the parabola in half.

35. 1) Quadratic Formulas and Their Graphs
The axis of symmetry is a line that divides the parabola in half.
The vertex is a maximum or minimum of the parabola.

36. 1) Quadratic Formulas and Their Graphs
The axis of symmetry here is x = 0
The vertex here is a minimum at (0, 0)

37. 1) Quadratic Formulas and Their Graphs
Points on the parabola have corresponding points that are equidistant from the axis of symmetry.
A B
A’ B’