Warm up
Put each of the following in slope intercept form
6x + 3y = 12
19 - y = 19x
5x + y - 2 = 12x – 3
y - 6x = 5
8x + y = 2x – 4
Review
• To graph a line when we have slope intercept
form, we can just plug in any value for x
– Example: y = 4x + 6
– Martha gets paid 12 dollars an hour and makes 10
dollars in tips every night. Make a table showing
how much she makes for working different
numbers of hours
2/27/13 Quadratic Functions
• I will be able to identify a quadratic function
and its vertex
• I will be able to find the maximum and
minimum of a quadratic function
Quadratic Functions
• A quadratic function is any function that can be
written in the form f(x)= ax2 + bx + c
– Remember a quadratic equation has a degree of ___
• The most basic quadratic is y = x2
– Lets plug in values for x and see what we get for y
Graph of a quadratic
y = x2
x
-4
-3
-2
-1
0
1
2
3
4
y
Try it
X=
2
4
6
10
f(x)= x2
f(x)= 2x2 + 3x – 1
Graph of a quadratic
• The graph of a quadratic function is called a
parabola.
– If a > 0, the parabola opens up
– If a < 0, the parabola opens down
Quadratic
Vertex of a quadratic
• The vertex refers to the highest or lowest point
of a parabola
– For a positive quadratic, the vertex is the lowest
point.
• The y-coordinate of the vertex is the minimum value of f.
– For a negative quadratic the vertex is the highest
point.
• The y-coordinate of the vertex is the maximum value of f.
Example
• Determine whether the following quadratic will
have a maximum or a minimum
a. f(x) = 5x2 + 3x – 9
b. g(x) = 12 + 8x – 12x2
c. f(x) = 4x2 - 9x + 3
d. g(x) = -6x + x2
Practice
a. f(x) = x2 + x – 6
b. g(x) = 5 + 4x – x2
c. f(x) = 2x2 - 5x + 2
d. g(x) = 7 - 6x - 2x2
Minimum and Maximums
• Enter the equation in your calculator
• Look at the graph
• Decide if you are finding a max or min
– If you can’t see a max or a min, change your
window
• Press 2nd TRACE and choose option
3:minimum or 4:maximum
• Use the left and right arrow keys to choose a
point to the left of the min or max and a point
to the right
Example 1
• Ex 2. Identify whether f(x) = -2x2 - 4x + 1 has a maximum value or a
minimum value at the vertex. Then give the approximate coordinates
of the vertex.
• First, graph the function:
• Next, find the maximum value of the parabola (2nd, Trace):
• Finally,
max(-1, 3).
Example
1. f(x) = -x2 + 6x – 8
Example
2. f(x) = 2x2 + 3x – 5
By hand
• If you don’t have a graphing calculator, the
minimum or maximum can be computed
using the following formula:
 b b 
 , f  
 2a 2a 
• As long as the equation is of the form:
y = ax2 + bx + c

Examples
• f(x) = -x2 + 6x – 8
• f(x) = 2x2 + 3x – 5
Think about it
• 1. a. Give an example of a quadratic function that has a
maximum value. How do you know that it has a
maximum?
• 1. b. Give an example of a quadratic function that has a
minimum value. How do you know that it has a
minimum?
Practice
• Find the maximum or minimum of each quadratic
f(x) = -5x2 + x – 4
f(x) = 3x - 2x2 – 2
f(x) = – 5 - 6x2 + 9x
f(x) = -6x2 + 2x – 9
f(x) = + 3x - x2 – 7
f(x) = 4x2 + 2x – 1