Chapter 9 9-1 Symmetry Definition: Symmetry Two points P and P 1 are symmetric with respect to a line l when they are the same distance from l, measured

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 #9.1Pg 389-390 1-33 Odd 35-62

Chapter 9

9-2 Translations

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.

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 9-6

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 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



1 3( ) . ( ) 1

2The graphof y f x is given Sketchthe graphof f x

x x

1 3( ) . ( ) 1

2The graphof y f x is given Sketchthe graphof f x

x x

( ) . ( ) 1 2The graphof y f x x is given Sketchthe graphof f x x

Find the function that is finally graphed after the series of transformations are applied to the graph of y x

2y x 3 2y x

c

d

c

a

HW# 9.3Pg 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 #9.4Pg 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 FunctionGraphing a Quadratic Function

Graph y = – (x + 3)2 + 412

To graph the function, first plot the vertex (h, k) = (– 3, 4).

(– 3, 4)SOLUTION The function is in vertex form

y = a (x – h)2 + k.

a = – , h = – 3, and k = 4

12

a < 0, the parabola opens down.

Graphing a Quadratic FunctionGraphing a Quadratic Function in Vertex Form

Use symmetry to completethe graph.

(– 3, 4)

(1, – 4)

(–1, 2)

(– 7, – 4)

(– 5, 2)

Graph y = – (x + 3)2 + 412

Draw the axis of symmetryx = – 3.

Plot two points on one side of it, such as (–1, 2) and (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 - 321

( 1) 22

y x


Goal 1 To analyze the graph of f(x) = ax2 + bx + c

Graphing a Quadratic Function

2Graph: ( 2) 1y x

Ex


Vocab


Vocab

2( ) 4 3f x x x 2( ) ( 2) 1f x x


Vocab

2 4 3y x x 2( 2) 1y x


Vocab

Definition

A quadratic function is a function that can be described as 2( ) ,where 0.f x ax bx c a

Definition

A quadratic function is a function that can be described as 2( ) ,where 0.f x ax bx c a

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


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?

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

Maximizing RevenueJoe 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 ProfitA 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 #9.6Pg 410-411 1-13 Odd, 24-43

Chapter 9

9-7 Graphs and x-intercepts

Goal 1To find the x-intercepts of the graph of a quadratic function

VERTEX AND INTERCEPT FORMS OF A QUADRATIC FUNCTION

FORM OF QUADRATIC FUNCTION CHARACTERISTICS OF GRAPH

Vertex form:

Intercept form:

y = a (x – h)2 + k

y = a (x – p )(x – q )

For both forms, the graph opens up if a > 0 and opens down if a < 0.

The vertex is (h, k ).

The axis of symmetry is x = h.

The x -intercepts are p and q.

The axis of symmetry is half-way between ( p , 0 ) and (q , 0 ).

THE GRAPH OF A QUADRATIC FUNCTIONCONCEPT

SUMMARY

• The axis of symmetry is the vertical line x = – .b2a

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 – .b2a

Graph y = 2 x 2 – 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.



Graph y = 2 x 2 – 8 x + 6

x = – = – = 2 b2 a

– 82(2)

y = 2(2)2 – 8 (2) + 6 = – 2

So, the vertex is (2, – 2).

(2, – 2)

The x-coordinate is:

The y-coordinate is:

Find and plot the vertex.

(2, – 2)


Graph y = 2 x 2 – 8 x + 6

Draw a parabola through the plotted points.

(0, 6)

(1, 0)

(4, 6)

(3, 0)

Draw the axis of symmetry x = 2.

Plot two points on one side of theaxis of symmetry, such as (1, 0)and (0, 6).

Use symmetry to plot two more points, such as (3, 0) and (4, 6).

Graphing a Quadratic Function in Intercept Form

Graph y = – ( x +2)(x – 4)

The quadratic function is in intercept form y = a (x – p)(x – q), where a = –1, p = – 2, and q = 4.

SOLUTION


Graph y = – ( x +2)(x – 4)

The axis of symmetry lies half-way between these points, at x = 1.

(– 2, 0) (4, 0)

The x-intercepts occur at (– 2, 0) and (4, 0).(– 2, 0) (4, 0)


Graph y = – ( x +2)(x – 4)

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)

(1, 9)

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 #9.7Pg 413 1-28

Chapter 9

9-8 Mathematical Modeling: Using Quadratic Functions

Goal 1To find a quadratic function given a graph or three 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)

2( ) 2 6 1f x x x

Find the quadratic function that fits the curve that contains the following three data points. (1, 6), (-2, 3), and (4, 18)

21 3( ) 4

2 2 f x x x

Find the quadratic function that fits the curve that contains the following three data points. (1, 4), (-1, 6), and (-2, 16)

2( ) 3 2 f x x x

Find the equation of the graph shown

21( ) ( 2) 1

2f x x

Find the equation of the graph shown

2( ) ( 3) 2 f x x

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 #9.8Pg 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-9Pg 424-426 1-38

Documents

Chapter 9 9-1 Symmetry Definition: Symmetry Two points P and P 1 are symmetric with respect to a line l when they are the same distance from l, measured