52 Objectives Graph quadratic functions Find the maximum and minimum value of quadratic functions By the end of today you should be able to Properties of Parabolas Recall The standard form of a quadratic function is ID: 432377
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Slide1
Properties of Parabolas
§
5.2Slide2
Objectives
Graph quadratic functions.
Find the maximum and minimum value of quadratic functions.
By the end of today, you should be able to…Slide3
Properties of Parabolas
Recall:
The
standard form of a quadratic function is
y = ax
2
+ bx +
c
When b = 0, the function simplifies
to
y
= ax
2
+
c
axis of symmetry:
x =
0
(the
y-
axis)
Vertex: (
0,c
)
(the y-intercept)Slide4
Graphing
a function in the form of y = ax
2
+ c
Step 1)
Graph the vertex
.
Step 2) Make a table of values to find some points on one side of the axis of symmetry x = 0.
Step 3) Graph corresponding points on the other side of the axis of symmetry.
Graph: y = -½x2 + 2
Step 4) Sketch the curve.Slide5
Graphing
a function in the form of y = ax
2
+ c
Step 1)
Graph the vertex
.
Step 2) Make a table of values to find some points on one side of the axis of symmetry x = 0.
Step 3) Graph corresponding points on the other side of the axis of symmetry.
Graph: y = 2x2 – 4
Step 4) Sketch the curve.Slide6
What about when b≠0?
How do we graph
y = ax
2
+ bx + c
when b ≠ 0
?
We know that a and
b both have an effect on the axis of symmetry.You can use the values of a, b and c to find characteristics of a graph of a quadratic function.Slide7
Key Concepts!
When
a > 0
, the parabola opens
up
.
When
a < 0
, the parabola opens down.
The axis of symmetry is the line . The x-coordinate of the vertex is .
The y-coordinate of the vertex is the value of y when , or . The y-intercept is
(0, c).Slide8
Graphing
a function in the form of y = ax
2
+ bx + c
Step 1) Find and graph the axis of symmetry.
Step 2) Find and graph the vertex
.
Step 3) Find and graph the y-intercept and its reflection
.
Graph: y = x
2 – 2x – 3Label the vertex and the axis of symmetry.
Step 4)
Evaluate the function for another value of x. Graph the point and its reflection
.
Step 5) Sketch the graph.Slide9
Graphing
a function in the form of y = ax
2
+ bx + c
Step 1) Find and graph the axis of symmetry.
Step 2) Find and graph the vertex
.
Step 3) Find and graph the y-intercept and its reflection
.
Graph: y = -x2
+ 4x +2Label the vertex and the axis of symmetry.
Step 4)
Evaluate the function for another value of x. Graph the point and its reflection
.
Step 5) Sketch the graph.Slide10
Finding a minimum value
Step 1) Decide which way the parabola opens
.
Step 2) Find the x-coordinate of the vertex
.
Step 3) Find the y-coordinate of the vertex.
Graph:
y = 3x
2
+ 12x + 8What is the minimum value of this function?
Step 4) Sketch the curve.Slide11
Finding a minimum value
Step 1) Decide which way the parabola opens
.
Step 2) Find the x-coordinate of the vertex
.
Step 3) Find the y-coordinate of the vertex.
Graph:
y = 2x
2
+ 8x – 1 What is the minimum value of this function?
Step 4) Sketch the curve.Slide12
The height of a football can be modeled with the quadratic function
h
= -0.01x
2
+ 1.18x + 2
.
The horizontal distance in feet from the point of impact with the kicker’s foot is
x, and h is the height of the ball in feet.What is the ball’s height when it has traveled 30 ft downfield
?What is the maximum height of the football? How far downfield has the ball traveled when it reaches its maximum height?Slide13
The equation for the cost in dollars of producing automobile tires
is
C = 0.000015x
2
– 0.03x + 35
, where
x
is the number of tires produced.
Find the number of tires that minimizes the cost.
What is the cost for that number of tires?Slide14
Ticketing
The manager of a symphony computes that the symphony will earn
-40P
2
+ 1100P
dollars per concert if they charge
P
dollars for tickets. What ticket price should the symphony charge in order to maximize its profits?Slide15
Real World Connection
The number of items a company sells frequently is a function of the item’s price (p). The revenue from sales of the item is the product of the price and the number sold.
Economics
The number of unicycles a company sells can be modeled by
-2.5p + 500
.
What price will maximize the company’s revenue from unicycle sales? What is the maximum revenue?Slide16
What are we being asked to do?
We know that
revenue
is
price
times the
number of unicycles sold
.
r =
revenue p = price of a unicycle.
-2.5p + 500 = number of unicycles sold.Write the equation:
Find the x-coordinate of the vertex:Then the y-coordinate of the vertex:
Find: What
price will maximize the company’s revenue from unicycle sales? What is the maximum revenue?Slide17
Homework
p.244 (10 – 30 even)
Don’t have
to graph!