Transcript Graphing Quadratic Functions

Graphing Quadratic Functions
Chapter 6.1
Quadratic Functions
Music managers handle publicity and other
business issues for the artists they manage.
One group’s manager has found that based on
past concerts, the predicted income for a
performance is
P ( x )   50 x  4000 x  7500
2
where x, is the price per ticket in dollars.
P ( x )   50 x  4000 x  7500
2
80
Income (thousands of dollars)
The graph of our functions is
shown here
Notice that at first the
income increases, as the
price per ticket increases,
but as the price continues
to increase, the income
declines.
How is this graph useful?
60
40
20
20
40
60
Price per ticket
80
Quadratic Functions
A quadratic function is described by an
equation of the following form:
f ( x )  ax  bx  c
2
The shape of the graph of any quadratic
function is called a parabola
One way to graph parabolas
If you have no idea how to graph a function,
your first game plan should be…?
PLOT POINTS!! PLOT
PLOT
POINTS!!
POINTS!!
PLOT POINTS!!
PLOT POINTS!!
Example 1
Plot points to graph the function:
f ( x)  x  3x  1
2
x
-3 -2 -1 0
1
f(x) -1 -3 -3 -1 3
Another way to graph quadratics

axis of symmetry
x = -b/2a
You need 3 things:
–
–
–
Axis of symmetry
Y-intercept
Vertex
f ( x )  ax  bx  c
2
vertex
y-intercept
(0, c)
f (2)   (2)  4(2)  1
4
b
 (4)
  8  1  2
f(  2 )   4
2
2a
2 (  1)
f (2)  5
2
Example 2

Find axis, vertex and
y-int to graph:
f ( x)   x  4 x  1
2
y-intercept: (0, 1)
axis: x = -2
vertex: (-2, 5)
?)
 bf
2a
Example 3

Find axis, vertex and
y-int to graph:
f ( x)  x
2
y-intercept: (0, 0)
axis: x = 0
vertex: (0, 0)
?)
( 0 )0 0

f ( 0 )2 (1) 0
2
0
f (0)  2(0)  1
b
0
2
f (0 )  0  1 0
2a
2 (1)
f (0)   1
Example 4

Find axis, vertex and
y-int to graph:
f ( x)  2 x  1
2
y-intercept: (0, -1)
axis: x = 0
vertex: (0, -1)
?)
f (2)   (2)  4(2)  3
b
4
f (  2)   4  
8  23
2 (  1)
2a
f (2)  7
2
Example 5

Find axis, vertex and
y-int to graph:
f ( x)   x  4 x  3
2
y-intercept: (0, 3)
axis: x = -2
vertex: (-2, 7)
?)
Maximum and minimum values

The y-coordinate of the vertex gives the
maximum or minimum value for a quadratic
function.
Maximum
a<0
Minimum
a>0
Vertex: (1, 4)
Example 1
f ( x)   x  2 x  3
2
• Does this function have a
maximum or a minimum value? (max)
• What is the max value? 4
f (1)   (1)  2 (1)  3
2
b
2a

2
2 (  1)

2
2
1
f (1)   1  2  3
f (1)  4
Example 2
A souvenir shop sells about 200 coffee mugs
each month for $6.00 each. The shop owner
estimates that for each $0.50 increase in the
price, he will sell about 10 fewer mugs per
month.
a.) How much should the owner charge for
each mug in order to maximize the monthly
income from their sales?
b.) What is the maximum income?
Example 2
In words:
Income equals the # of mugs sold multiplied
by the price per mug.
In variables:
Let x = the number of $0.50 price increases
Then price is… 6  0 . 5 x
And # sold is… 200  10 x
Example 2
Equation: Income = Mugs x Price
I ( x )  ( 200  10 x ) ( 6  0 . 5 x )
I ( x )  1200  100 x  60 x  5 x
I ( x )   5 x  40 x  1200
2
2