1. Graphs of Quadratic Functions

2. Quadratic Functions:
Definition of Quadratic Function:
The graph of a quadratic function is a special type of “U”-shaped
curve that is called a parabola. (U shaped)
All parabolas are symmetric with respect to a line called the
axis of symmetry, or simply the axis of the parabola.
The point where the axis intersects the parabola is the vertex of the
parabola.

3. Quadratic Equations y = ax2 + bx + c y = a(x - h)2 + k f(x) =
Standard form
Vertex form
(later)
The graph is “U-shaped” and is called a parabola.

4. The highest or lowest point on the parabola is called the vertex.

5. The axis of symmetry for the parabola is the vertical line through the vertex.

6. y = x2 y = -x2 Vertex is a minimum Vertex is a maximum Opens up
Opens down

7. Finding the vertex of a parabola:
This gives you the
x-value and plug the
x-value into the original
function to find the
y-value

8. EXAMPLES:
Sketch the graph of the quadratic function without using a graphing
utility. Identify the vertex and x-intercepts.
You know the graph will go up b/c
a is a positive number
Find the vertex first!!
So the x part of the vertex
is -2
Now, to find the y-value,
plug in -2 wherever you
see an x.
Vertex: (-2,-7)

9. Now make the table! Remember the vertex goes in the middle!!! (x, y) x-2

10. Sketch the graph of the quadratic function without using a graphing
EXAMPLES:
Sketch the graph of the quadratic function without using a graphing
utility. Identify the vertex and x-intercepts.
You know the graph will go up b/c
a is a positive number
Find the vertex first!!
Where is
B???
So the x part of the vertex
is 0
Now, to find the y-value,
plug in zero wherever you
see an x.
Vertex: (0,-5)

11. So, we know that the vertex is (0,-5) now all we have to do is a
T-chart and pick 4 more points and then graph.
x y
0 -5

12. One More….
Name the vertex, axis of symmetry and whether it opens up or down
AND GRAPH.
y = 2x2 + 1
Vertex: (0, 1)
a = 2
b = 0
c = 1
Axis of sym: x = 0
Opens up

13. Name the vertex, axis of symmetry and whether it opens up or down
AND GRAPH.
y = 2x2 + 1
Vertex: (0, 1) x y
Put vertex in the middle of the t-table -2 9 -1 3 1 1 3 2 9

14. Example Graph the quadratic function f (x) = -x2 + 6x -. Solution:
Step 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.
Step Find the vertex. We know the x-coordinate of the vertex is –b/2a.
We 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:
the parabola has its vertex at (3,7).

15. Example Graph the quadratic function f (x) = -x2 + 6x -.
Step Find the x-intercepts. Replace f (x) with 0 in f (x) = -x2 + 6x = -x2 + 6x - 2

16. Example Graph the quadratic function f (x) = -x2 + 6x -.
Step 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).
Step Graph the parabola.

17. 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)).

18. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions
Read the problem carefully and decide which quantity is to be maximized or minimized.
Use the conditions of the problem to express the quantity as a function in one variable.
Rewrite the function in the form f(x) = ax2 + bx +c.
Calculate -b/(2a). 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)).
Answer the question posed in the problem.