Chapter 9
9-1 Symmetry
Definition: Symmetry
Two points P and P1 are symmetric with respect to a
line l when they are the same distance from l,
measured along a perpendicular to l. Line l is known
as a line or axis of symmetry. P1 is said to be the
image of P.
A figure, or set of points, is symmetric with respect to
a line when the image of each point in the set is also
in the set.
Theorem 9-1
Two points are symmetric with respect to the x-axis if
and only if their y-coordinates are additive inverses and
they have the same x-coordinate
Two points are symmetric with respect to the y-axis if
and only if their x coordinates are additive inverses of
each other and they have the same y value
Theorem 9-2: When a relation if defined by an equation
A. its graph is symmetric to the y-axis if and only if
replacing x by –x produces an equivalent equation
B. its graph is symmetric with respect to the x-axis if and
only if replacing y by –y produces an equivalent equation
Definition: Symmetry with respect to a point
Two points P and P1 are symmetric with respect to a
point Q when they are the same distance from Q, and
all three points are collinear. P1 is said to be the
image of P.
A figure, or set of points, is symmetric with respect to
a point when the image of each point in the set is also
in the set.
Theorem 9-3
Two points are symmetric to the origin if and only if both
their x- and y-coordinates are additive inverses of each
other
Theorem 9-4
A graph of a relation defined by an equation is symmetric
with respect to the origin if and only if replacing x by –x
and y by –y produces an equivalent equation
Definition
A function is an even function when f(-x) = f(x) for all x in
the domain of f.
Definition
A function is an odd function when f(-x) = -f(x) for all x in
the domain of f.
Determine if the function is odd even or neither
HW #9.1
Pg 389-390 1-33 Odd 35-62
Chapter 9
9-2 Translations
An alteration of a relation is called a transformation. Moving the
graph of a relation without changing the shape or size is called a
translation
Theorem 9-5
In an equation of a relation, replacing y by y – k,
where k is a constant, translates the the graph
vertically a distance of |k|.
If k is positive, the translation is up.
If k is negative, the translation is down.
Theorem 9-6
In an equation of a relation, replacing x by x – h,
where h is a constant, translates the graph
horizontally a distance of |h|.
If h is positive, the translation is right.
If h is negative, the translation is left.
Consider the graph of y = |x|, sketch the graphs of the
following by translating
Given the graph of y = x2 use transformations to sketch the
graph of y = (x + 3)2 - 5
The graph of a function is given, sketch the graph under the
given translations
(c) Q(x) = f(x – 3) + 2
HW #9.2
Pg 393 1-28, 29-39 Odd
Chapter 9
9-3 Stretching and Shrinking
Goal 1 To sketch a graph showing vertical stretching or
shrinking
Goal 2 To sketch a graph showing horizontal stretching or
shrinking
Multiplying the function causes the graph to stretch or shrink
vertically
The negative causes a reflection over the x-axis
The negative causes a reflection over the x-axis
The negative causes a reflection over the x-axis
The graph of f(x) is given use the graph to find the graph of y = 2f(x)
The graph of y = 2f(x) is
obtained by multiplying each
y-value by 2
To get the same y-values, the x-values must be smaller as
in Y2 larger as in Y3
Opposite x-values have the same y-values
Opposite x-values have the same y-values
Opposite x-values have the same y-values
1
3
The graph of y  f ( x)  is given. Sketchthe graph of f ( x) 
1
x
x2
1
3
The graph of y  f ( x)  is given. Sketchthe graph of f ( x) 
1
x
x2
The graph of y  f ( x)  x is given. Sketchthe graph of f ( x )  1  x  2
Find the function that is finally graphed after the series of
transformations are applied to the graph of y  x
y   x  2
y   x32
c
d
c
a
HW# 9.3
Pg 398-399 1-25 Odd, 26-53
9.4
A quadratic function has the form
y = ax 2 + bx + c where a  0.
The graph is “U-shaped” and is called
a parabola.
The highest or lowest point on the
parabola is called the vertex.
In general, the axis of symmetry for
the parabola is the vertical line
through the vertex.
Sketch the graph of the following find the vertex, line of symmetry
and the maximum or minimum value:
30. y < x2
31. y  x2
32. y  -3(x + 3)2
HW #9.4
Pg 402-403 1-25 Odd, 26-29
Chapter 9
9-5 Graphs of Quadratic Functions
Goal 1 To analyze the graph of f(x) = a(x – h)2 + k
Graphing a Quadratic Function
Graph y = –
1
2
(x + 3)2 + 4
SOLUTION
(– 3, 4)
The function is in vertex form
y = a (x – h)2 + k.
a = – 1 , h = – 3, and k = 4
2
a < 0, the parabola opens down.
To graph the function, first plot the vertex (h, k) = (– 3, 4).
Graphing a Quadratic Function in Vertex Form
Graph y = –
1
2
(x + 3)2 + 4
Draw the axis of symmetry
x = – 3.
(– 3, 4)
(– 5, 2)
(–1, 2)
Plot two points on one side of
it, such as (–1, 2) and (1, – 4).
Use symmetry to complete
the graph.
(– 7, – 4)
(1, – 4)
Sketch the graph of the following find the vertex, line of symmetry
and the maximum or minimum value:
Without graphing, find the vertex, line of symmetry, and the
minimum or maximum value.
1. F(x) = (x – 5)2 + 40
2. F(x) = -3(x – 5)2
3. F(x) = 2(x + 3)2 - 6
4. F(x) = -4(x + 9)2 + 3
Write the equation of the parabola that is a transformation of
f(x) = 2x2 and has a minimum or maximum value at the given
point
1. Maximum (0, 5)
2. Minimum (3, 0)
3. Minimum (3, -2)
4. Maximum (-2, 5)
HW #9.5
Pg 406-407 1-21 Odd, 22-27
HW Quiz #9.5
1) Graph
y = 4(x + 3)2 - 5
2) Graph
y = -5(x - 2)2 + 4
3) Graph
4) Graph
y = (x + 2)2 - 3
1
y  ( x  1) 2  2
2
9-6 Graphs of Quadratic Functions
Goal 1 To analyze the graph of f(x) = ax2 + bx + c
Graphing a Quadratic Function
Graph: y  ( x  2) 2  1
Graphing a Quadratic Function
Graphing a Quadratic Function
f ( x)  x  4 x  3
2
f ( x)  ( x  2)  1
2

Graphing a Quadratic Function
y  x  4x  3
2
y  ( x  2)  1
2

Graphing a Quadratic Function

Definition
A quadratic function is a function that can be described as
f ( x)  ax 2  bx  c, where a  0.
Definition
A quadratic function is a function that can be described as
f ( x)  ax 2  bx  c, where a  0.
To find the x-coordinate of the vertex, substitute 1 for a
and -4 for b in the formula:
To find the y-coordinate of the vertex, substitute 2 for x in
the original equation, and solve for y.
The vertex is (2, -1)
Graph the quadratic function.
Chapter 9
9-6 Graphs of Quadratic Functions
Goal 1 To find the standard form of a quadratic equation
Goal 2 To solve maximum and minimum problems
Definition
Three Forms of a quadratic equation
1. Definition: f(x) = ax2 + bx + c
2. Vertex Form: f(x) = a(x – h)2 + k
•
The book refers to this as Standard Form
3. Intercept Form: f(x) = a(x – p)(x – q)
•
p and q are the x-intercepts of the graph
To Convert from the definition form to standard/vertex form, we
complete the square.
Write the quadratic in standard/Vertex form and sketch the graph.
A rancher is fencing off a rectangular area with a fixed perimeter of 76
m. What dimensions would yield the maximum area? Find the
maximum area.
Beth has 3000 feet of fencing available to enclose a rectangular field.
One side of the field lies along a river, so only three sides require
fencing. Find the dimensions of the pen that will yield the maximum
area. What is the maximum area?
Maximizing Revenue
A publisher thinks that she can sell 1000 copies of a book on
sailing priced at $20 per copy. For each dollar she lowers the
price she thinks she can sell an additional 500 copies. According
to her thinking, at what price will her total revenue from sales be a
maximum, how many copies should she expect to sell at this
price, and what would be the expected revenue?
Maximizing Revenue
Joe Blow owns a commuter airline transport business. He transports
about 800 passengers a day between Chicago and Fort Wayne. A
round-trip ticket is $300. Joe has figured out that for every $5.00
increase in the ticket price, 10 passengers would be lost to the
competition. What ticket price should Joe charge to maximize his
income and what would his maximum income be?
Maximizing Profit
A 300 room hotel in Las Vegas is filled to capacity every night at
$80 a room. For each $1 increase in rent, 3 fewer rooms are
rented. If each room rented costs $10 to service per day, how
much should the management charge for each room to maximize
gross profit? What is the gross profit?
HW #9.6
Pg 410-411 1-13 Odd, 24-43
Chapter 9
9-7 Graphs and x-intercepts
To find the x-intercepts of the graph of a quadratic
Goal 1 function
VERTEX AND INTERCEPT FORMS OF A QUADRATIC FUNCTION
FORM OF QUADRATIC FUNCTION
CHARACTERISTICS OF GRAPH
Vertex form: y = a (x – h)2 + k
The vertex is (h, k ).
The axis of symmetry is x = h.
Intercept form: y = a (x – p )(x – q ) The x -intercepts are p and q.
The axis of symmetry is halfway between ( p , 0 ) and (q , 0 ).
For both forms, the graph opens up if a > 0 and opens down if a < 0.
CONCEPT
THE GRAPH OF A QUADRATIC FUNCTION
SUMMARY
The graph of y = a x 2 + b x + c is a parabola with these
characteristics:
• The parabola opens up if a > 0 and opens down if a < 0.
The parabola is wider than the graph of y = x 2 if a < 1 and
narrower than the graph y = x 2 if a > 1.
• The x-coordinate of the vertex is – b .
2a
b
• The axis of symmetry is the vertical line x = – 2a .
Graphing a Quadratic Function
2
Graph y = 2 x – 8 x + 6
SOLUTION
Note that the coefficients for this function
are a = 2, b = – 8, and c = 6.
Since a > 0, the parabola opens up.
Graphing a Quadratic Function
2
Graph y = 2 x – 8 x + 6
Find and plot the vertex.
The x-coordinate is:
x = – b = –– 8 = 2
2(2)
2a
The y-coordinate is:
y = 2(2)2 – 8 (2) + 6 = – 2
So, the vertex is (2, – 2).
(2, – 2)
Graphing a Quadratic Function
2
Graph y = 2 x – 8 x + 6
Draw the axis of symmetry x = 2.
(4, 6)
(0, 6)
Plot two points on one side of the
axis of symmetry, such as (1, 0)
and (0, 6).
Use symmetry to plot two more
points, such as (3, 0) and (4, 6).
Draw a parabola through the plotted
points.
(1, 0)
(3, 0)
(2, – 2)
Graphing a Quadratic Function in Intercept Form
Graph y = – ( x +2)(x – 4)
SOLUTION
The quadratic function is in intercept
form y = a (x – p)(x – q), where
a = –1, p = – 2, and q = 4.
Graphing a Quadratic Function in Intercept Form
Graph y = – ( x +2)(x – 4)
The x-intercepts occur at
(– 2,
2, 0)
0) and (4,
(4, 0).
The axis of symmetry lies half-way
between these points, at x = 1.
(– 2, 0)
(4, 0)
Graphing a Quadratic Function in Intercept Form
Graph y = – (x +2)(x – 4)
(1, 9)
So, the x-coordinate of the vertex is
x = 1 and the y-coordinate of the
vertex is:
y = – (1 + 2)(1 – 4) = 9
(– 2, 0)
(4, 0)
Graph the quadratic function. Label the vertex, axis of
symmetry, y-intercepts and y-intercepts.
b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
HW #9.7
Pg 413 1-28
Chapter 9
9-8 Mathematical Modeling: Using
Quadratic Functions
To find a quadratic function given a graph or three
Goal 1 data points
Goal 2 To solve problems using quadratic functions
Find the quadratic function that fits the curve that contains
the following three data points. (-2, 5), (-1, 5), and (0, 1)
f ( x)  2 x 2  6 x  1
Find the quadratic function that fits the curve that contains
the following three data points. (1, 6), (-2, 3), and (4, 18)
1
3
f ( x)  x 2  x  4
2
2
Find the quadratic function that fits the curve that contains
the following three data points. (1, 4), (-1, 6), and (-2, 16)
f ( x)  3 x 2  x  2
Find the equation of the graph shown
1
f ( x)  ( x  2) 2  1
2
Find the equation of the graph shown
f ( x)  ( x  3)2  2
A rocket is fired upward. At the end of the burn it has an
upward velocity of 245 m/s and is 14.1 meters high. Find the
maximum height it attains and at what time will it hit the
ground?
It will be 3077 m high when t = 25 seconds
It will hit the ground after 50.1 seconds
HW #9.8
Pg 418-419 1-27
Test Review
•
•
•
•
•
•
Transformations as applied to x2 or |x| or f(x)
Symmetry over y-axis/x-axis/origin
Odd/Even Functions
Find the equation of a parabola
Graph Parabolas and Absolute Value
Max/Min
A long rectangular sheet of metal,
12 inches wide, is to be made into a
rain gutter by turning up two sides
so that they are perpendicular to the
sheet. How many inches should be
turned up to give the gutter its
greatest capacity?
A person standing on the top of a building projects an object directly
upward with a velocity of 144ft/sec. Its height s(t) in feet above the
ground after t seconds is given by s(t) = -16t2 + 144t + 100.
a)What is its maximum height?
b)What is the height of the building?
c)What is the initial velocity?
A rocket is shot straight up into the air with an initial velocity of Vo ft/sec,
and its height s(t) in feet above the ground after t seconds is given by s(t)
= -16t2 + Vot.
(a)The rocket hits the ground after 12 seconds. What is its initial
velocity Vo?
(b)What is the maximum height attained by the rocket?
A cable television firm presently serves 5000 households and charges
$20 per month. A marketing survey indicates that each decrease of $1 in
the monthly charge will result in 500 new customers. Let R(x) denote the
total monthly revenue when the monthly charge is x dollars.
(a)Determine the revenue function R.
(b)Sketch the graph of R
(c)Find the value of x that results in maximum monthly revenue.
One thousand feet of chain-link fence is to be used to construct six cages
for a zoo exhibit. The design is shown in the figure.
(a) Express the width y as a function of the length x.
(b) Express the total enclosed area A of the exhibit as a function of x.
(c) Find the dimensions that maximize the enclosed area.
1. If a continuous function is even and the points
(-2, 4), (1, 2) and (-5, 10) are on the graph,
name three more points that are on the graph.
•
(2, 4), (-1, 2), (5, 10)
2. If a continuous function is odd and the points
(-2, 4), (1, 2) and (-5, 10) are on the graph,
name three more points that are on the graph.
•
(2, -4), (-1, -2), (5, -10)
1. If f(x) contains the point (-2, 4) and g(x) is a
transformation of f(x) such that g(x) = 3f(x-2) + 4.
What point on g(x) corresponds to the point of
(-2, 4) on f(x)?
•
(0, 16)
2. If g(x) contains the point (-2, 16) and g(x) is a
transformation of f(x) such that g(x) = 3f(x-2) + 4.
What point on f(x) corresponds to the point of
(-2, 16) on g(x)?
•
(-4, 4)
A rectangle is drawn such that one of its vertices is on the origin and
the other is on the graph of y = -x2 + 4. Find the dimensions for x and
y that maximize the area of the rectangle.
HW #R-9
Pg 424-426 1-38