1. Quadratic Functions

2. Graphs of Quadratic Functions
A Parabola.

3. The Quadratic Function
A quadratic function can be expressed in three formats:
f(x) = ax2 + bx + c is called the standard form
is called the factored form
f (x) = a(x - h)2 + k, is called the vertex form

4. The Standard Form f(x)=ax2 + bx + c If a > 0 then parabola opens up
If a < 0 then parabola opens down
(0,c) is the y-intercept
Axis of symmetry is

5. The Vertex Form f (x) = a(x - h)2 + k, a  0
***y – k = a(x-h)2
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric to the line x = h
If a > 0, the parabola opens upward
if a < 0, the parabola opens downward

6. Graphing Parabolas-Vertex Form
To graph f (x) = a(x - h)2 + k:
Determine whether the parabola opens up or down.
If a > 0, it opens up.
If a < 0, it opens down.
Determine the vertex of the parabola--- V(h, k).
Find any x-intercepts.
Replace f (x) with 0.
Solve the resulting quadratic equation for x.
Find the y-intercept by replacing x with zero.
Plot the intercepts and vertex. Connect these points with a smooth curve that is shaped like a cup.

7. The Factored Form Factored Form of a quadratic
r represents roots or x intercepts
Axis of symmetry:

8. Vertex form f (x) = a(x - h)2 + k
Example
Graph the quadratic function f (x) = -2(x - 3)2 + 8.
Vertex form f (x) = a(x - h)2 + k  
a=-2 h =3 k = 8
Given equation f (x) = -2(x - 3)2 + 8
This is a parabola that opens down with vertex V(3,8)

9. Example cont. The x-intercepts are 1 and 5.
Find the vertex. The vertex of the parabola is at (h, k). Because h = 3 and k = 8, the parabola has its vertex at (3, 8).
Find the x-intercepts. Replace f (x) with 0: f (x) = -2(x - 3)2 + 8.
0 = -2(x - 3) Find x-intercepts, setting f (x) equal to zero.
2(x - 3)2 = 8
(x - 3)2 = Divide both sides by 2.
(x - 3) = ± Apply the square root method.
x - 3 = or x - 3 = Express as two separate equations.
x = or x = Add 3 to both sides in each equation.
The x-intercepts are 1 and 5.
The parabola passes through (1, 0) and (5, 0).

10. Example cont. Graph the parabola. With a vertex at (3, 8)
f (0) = -2(0 - 3) = -2(-3) = -2(9) + 8 = -10
Find the y-intercept. Replace x with 0 in f (x) = -2(x - 3)2 + 8.
The y-intercept is –10. The parabola passes through (0, -10).
Graph the parabola.
With a vertex at (3, 8)
x-intercepts at (1,0) and (5,0)
and a y-intercept at (0,–10)
the axis of symmetry is the vertical line x = 3.

11. The Vertex of a Parabola Whose Equation Is f (x) = ax 2 + bx + c
Consider the parabola defined by the quadratic function
f (x) = ax 2 + bx + c. The parabola's vertex is at

12. Example Graph the quadratic function f (x) = -x2 + 6x -.
Determine how the parabola opens.
Note that a, the coefficient of x 2, is -1. Thus, a < 0;
This negative value tells us that the parabola opens downward.
Find the vertex.
x = -b/(2a).
Identify a, b, and c to substitute the values into the equation for the x-coordinate:
x = -b/(2a) = -6/2(-1)=3.
The x-coordinate of the vertex is 3.
We substitute 3 for x in the equation of the function to find the y-coordinate:
y=f(3) = -(3)^2+6(3)-2= =7,
the parabola has its vertex at (3,7).

13. Example Graph the quadratic function f (x) = -x2 + 6x -.
Find the x-intercepts. Replace f (x) with 0 in f (x) = -x2 + 6x - 2.
0 = -x2 + 6x (If you cannot factor)

14. Example Graph the quadratic function f (x) = -x2 + 6x -.
Find the y-intercept. Replace x with 0 in f (x) = -x2 + 6x - 2.
f (0) = • = -
The y-intercept is –2. The parabola passes through (0, -2).
Graph the parabola.

15. Minimum and Maximum: Quadratic Functions
Consider f(x) = ax2 + bx +c.
If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)).
If a < 0, the f has a maximum that occurs at x = -b/(2a). This maximum value is f(-b/(2a)).

16. The Discriminant and the Kinds of Solutions to ax2 + bx +c = 0
No x-intercepts
No real solution;
two complex imaginary solutions
b2 – 4ac < 0
One x-intercept
One real solution
(a repeated solution)
b2 – 4ac = 0
Two x-intercepts
Two unequal real solutions
b2 – 4ac > 0
Graph of
y = ax2 + bx + c
Kinds of solutions
to ax2 + bx + c = 0
Discriminant
b2 – 4ac

17. Quadratic Functions