chapter 1 - foundations for functions algebra ii

Chapter 1 - Foundations for Functions

Algebra II

Table of Contents

• 1.6- Relations and Functions• 1.7 - Function Notation• 1.8 - Exploring Transformations• 1.9 - Introduction to Parent Functions

1.6 - Relations and Functions

Algebra II

A relation is a pairing of input values with output values.

It can be shown as a set of ordered pairs (x,y), where x is an input and y is an output.

The set of input values for a relation is called the domain, and the set of output values is called the range.

1. Define the words below

1-6 Algebra II (bell work)

A

B

C

2

Domain Range

Mapping Diagram

Set of Ordered Pairs

{(2, A), (2, B), (2, C)}

(x, y) (input, output) (domain, range)

1-6

Give the domain and range for this relation:

{(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)}

{(100,5), (120,5), (140,6), (160,6), (180,12)}.

Domain: {100, 120, 140, 160, 180}

Range: {5, 6, 12}

List the set of ordered pairs:

1-6 Example 1 Identifying Domain and Range

Give the domain and range for the relation shown in the graph.

Domain: {–2, –1, 0, 1, 2, 3}

Range: {–3, –2, –1, 0, 1, 2}

List the set of ordered pairs:

{(–2, 2), (–1, 1), (0, 0), (1, –1), (2, –2), (3, –3)}

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A relation in which the first coordinate is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range.

Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C.

Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range.

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Example 2 Determining Whether a Relation is a Function

Determine whether each relation is a function.A. from the items in a store to their prices on

a certain date

B. from types of fruits to their colors

There is only one price for each different item on a certain date. The relation from items to price makes it a function.

A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function.

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Determine whether each relation is a function.

B. from the number of items in a grocery cart to the total cost of the items in the cart

There is only one price for each shoe size. The relation from shoe sizes to price makes is a function.

The number items in a grocery cart would be associated with many different total costs of the items in the cart. The relation of the number of items in a grocery cart to the total cost of the items is not a function.

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Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.

This is a function. Any vertical line would pass through only one point on the graph.

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Example 3 Using the Vertical-Line Test

This is not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2).

Use the vertical-line test to determine whether the relation is a function.

If not, identify two points a vertical line would pass through.

1-6



This is a function. Any vertical line would pass through only one point on the graph.

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This is not a function. A vertical line at x = 1 would pass through (1, 2) and (1, –2).



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HW pg. 47• 1.6-

– 9 - 16, 18, 30, 31-37, 39, 40, 43, 51-53, 58-61• Do not need to copy down graphs and charts

– HW Guidelines or ½ off– Always staple Day 1&2 Together– Put assignment in planner

1.7- Function Notation

Algebra II

ƒ(x) = 5x + 3 ƒ(1) = 5(1) + 3

Output value Input valueInput value

ƒ of x equals 5 times x plus 3. ƒ of 1 equals 5 times 1 plus 3.

Function Notation


The function described by ƒ(x) = 5x + 3 is the same as the function described by y = 5x + 3.

And both of these functions are the same as the set of ordered pairs (x, 5x+ 3).

y = 5x + 3 (x, y) (x, 5x + 3)

ƒ(x) = 5x + 3 (x, ƒ(x)) (x, 5x + 3)

Notice that y = ƒ(x) for each x.

The graph of a function is a picture of the function’s ordered pairs.

1-7

f(x) is not “f times x” or “f multiplied by x.” f(x) means “the value of f at x.” So f(1) represents the value of f at x =1

Caution

ƒ(x) = 8 + 4x Substitute each value for x and evaluate.

For each function, evaluate ƒ(0), ƒ , and ƒ(–2).

ƒ(0) = 8 + 4(0) = 8

ƒ(–2) = 8 + 4(–2) = 0

ƒ = 8 + 4 = 10

1-7 Example 1 Evaluating Functions


Use the graph to find the corresponding y-value for each x-value.

ƒ(0) = 3

ƒ = 0

ƒ(–2) = 4

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ƒ(x) = x2 – 4x

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The output ƒ(x) of a function is called the dependent variable because it depends on the input value of the function.

The input x is called the independent variable. When a function is graphed, the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

1-7

Math Joke

• Parent: Did you study your algebra lesson at the family reunion?

• Student: Sure, it was a function with relations?

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Graph the function f(x) = 3x – 1.

x 3x – 1 f(x)

– 1 3(– 1) – 1 – 4

0 3(0) – 1 – 1

1 3(1) – 1 2

Make a table. Graph the points.

1-7 Example 2 Graphing Functions

Graph the function.

Graph the points.

Do not connect the points because the values between the given points have not been defined.

3 5 7 9

2 6 10

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A carnival charges a $5 entrance fee and $2 per ride.Write a function to represent the total cost after taking a certain number of rides.

Let r be the number of rides and let C be the total cost in dollars. The entrance fee is constant.

C(r) = 5 + 2r

First, identify the independent and dependent variables.

Cost depends on the entrance fee plus the number of rides taken

Cost = entrance fee + number of rides taken

Replace the words with expressions.

Dependent variable Independent variable

1-7 Example 3 Application

What is the value of the function for an input of 12, and what does it represent?

Substitute 12 for r and simplify.

The value of the function for an input of 12 is 29. This means that it costs $29 to enter the carnival and take 12 rides.

C(12) = 5 + 2(12)

C(12) = 29

A carnival charges a $5 entrance fee and $2 per ride.

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A local photo shop will develop and print the photos from a disposable camera for $0.27 per print.

Write a function to represent the cost of photo processing.

Let x be the number of photos and let f be the total cost of the photo processing in dollars.

f(x) = 0.27x

First, identify the independent and dependent variables.Cost depends on the number of photos processed

Cost = 0.27 number of photos processed

Replace the words with expressions.

Dependent variable Independent variable

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Substitute 24 of x and simplify.f(24) = 0.27(24)

= 6.48

What is the value of the function for an input of 24, and what does it represent?

The value of the function for an input of 24 is 6.48. This means that it costs $6.48 to develop 24 photos.

A local photo shop will develop and print the photos from a disposable camera for $0.27 per print.

1-7

HW pg. 54

• 1.7– 11-22 (Don’t copy graphs), 33-37, 40, 45-48 (No

Graphs), 59, 66– Ch: 38

1.8 - Exploring Transformations

Algebra II

A transformation is a change in the position, size, or shape of a figure.

A translation, or slide, is a transformation that moves each point in a figure the same distance in the same direction.


Review notes, notebooks, hw sheets, checking odd answers, HWG

Perform the given translation on the point (–3, 4). Give the coordinates of the translated point.

5 units right

Translating (–3, 4) 5 unitsright results in the point (2, 4).

(2, 4)

5 units right(-3, 4)

Example 1 Translating Points1.8

2 units left and 2 units down

Translating (–3, 4) 2 unitsleft and 2 units down resultsin the point (–5, 2).

(–3, 4)

(–5, 2)

2 units

3 units


1.8

4 units right


Translating (–1, 3) 4 unitsright results in the point (3, 3).

(–1, 3)

4 units

(3, 3)

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1 unit left and 2 units down


Translating (–1, 3) 1 unit left and 2 units down resultsin the point (–2, 1).

(–1, 3)

(–2, 1)

1 unit

2 units

1-8

Notice that when you translate left or right, the x-coordinate changes, and when you translate up or down, the y-coordinate changes.

TranslationsHorizontal Translation Vertical Translation

1-8

Do not copy translations chart

ReflectionsReflection Across y-axis Reflection Across x-axis

1-8

Use a table to perform each transformation of y=f(x). Use the same coordinate plane as the original function.

translation 2 units up

x y y + 2–5 –3 –3 + 2 = –1

–2 0 0 + 2 = 2

0 –2 –2 + 2 = 0

2 0 0 + 2 = 2

5 –3 –3 + 2 = –1

1-8 Example 2 Translating and Reflecting Functions

reflection across x-axis

Identify important points from the graph and make a table.

x y –y–5 –3 –1(–3) = 3

–2 0 – 1(0) = 0

0 –2 – 1(–2) = 2

2 0 – 1(0) = 0

5 –3 – 1(–3) = 3

1-8

Math Joke

• Student: I had French class today. I’ve done enough Algebra!

• Parent: How?

• Student: All we did was translate!

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translation 3 units right

Use a table to perform the transformation of y = f(x). Use the same coordinate plane as the original function.

x y x + 3

–2 4 –2 + 3 = 1

–1 0 –1 + 3 = 2

0 2 0 + 3 = 3

2 2 2 + 3 = 5

1-8 Just Watch

reflection across x-axis

x y –y

–2 4 –4

–1 0 0

0 2 –2

2 2 –2

f

Use a table to perform the transformation of y = f(x).

Use the same coordinate plane as the original function.

1-8 Just Watch

Stretches and Compressions1-8 Day 2

v

Use a table to perform a horizontal stretch of the function y = f(x) by a factor of 3.

Graph the function and the transformation on the same coordinate plane.


3x x y3(–1) = –3 –1 3

3(0) = 0 0 0

3(2) = 6 2 2

3(4) = 12 4 2

1-8 Example 3 Stretching and Compressing Functions


Use a table to perform a vertical stretch of y = f(x) by a factor of 2.

Graph the transformed function on the same coordinate plane as the original figure.

x y 2y–1 3 2(3) = 6

0 0 2(0) = 0

2 2 2(2) = 4

4 2 2(2) = 4

1-8

The graph shows the cost of painting based on the number of cans of paint used.

Sketch a graph to represent the cost of a can of paint doubling, and identify the transformation of the original graph that it represents.

If the cost of painting is based on the number of cans of paint used and the cost of a can of paint doubles, the cost of painting also doubles. This represents a vertical stretch by a factor of 2.

1-8 Example 4 Business Application

What if…? Suppose that a discounted rate is of the original rate.

Sketch a graph to represent the situation and identify the transformation of the original graph that it represents.

If the price is discounted by of the hourly rate, the value of each y-coordinate would be multiplied by .

1-8

HW pg. 63

• 1.8– Day 1: 14-20 (Sketch both on one), 59, 61– Day 2: 21-27 (Sketch both on one), 37, 44, 45– Ch: 36

1.9 - Introduction to Parent Functions

Algebra II

1. Copy the chart below

Parent Functions

1-9 Algebra 2 (bell work)

Identify the parent function for g from its function rule.

Then graph g on your calculator and describe what transformation of the parent function it represents.

g(x) = x – 3

Graph Y1 = x – 3 on the graphing calculator. The function g(x) = x – 3 intersects the y-axis at the point (0, –3).

g(x) = x – 3 is linear

So g(x) = x – 3 represents a vertical translation of the linear parent function 3 units down.

The linear parent function ƒ(x) = x intersects the y-axis at the point (0, 0).

1-9 Example 1 Identifying Transformations of Parent Functions


Then graph on your calculator and describe what transformation of the parent function it represents.

g(x) = x2 + 5

Graph Y1 = x2 + 5 on a graphing calculator. The function g(x) = x2 + 5 intersects the y-axis at the point (0, 5).

g(x) = x2 + 5 is quadratic.

So g(x) = x2 + 5 represents a vertical translation of the quadratic parent function 5 units up.

The quadratic parent function ƒ(x) = x intersects the y-axis at the point (0, 0).

1-9

g(x) = x3 + 2

g(x) = x3 + 2 is cubic.

The cubic parent function ƒ(x) = x intersects the y-axis at the point (0, 0).

Graph Y1 = x3 + 2 on a graphing calculator. The function g(x) = x3 + 2 intersects the y-axis at the point (0, 2).

So g(x) = x3 + 2 represents a vertical translation of the quadratic parent function 2 units up.


Then graph on your calculator and describe what transformation of the parent function it represents.

1-9

Math Joke

• Teacher: Why did your mother and father do your algebra homework?

• Student: They really understand parent functions

1-9

Graph the data from this set of ordered pairs.

Describe the parent function and the transformation that best approximates the data set.

{(–2, 12), (–1, 3), (0, 0), (1, 3), (2, 12)}

x –2 –1 0 1 2

y 12 3 0 3 12

The graph of the data points resembles the shape of the quadratic parent function ƒ(x) = x2.

The quadratic parent function passes through the points (1, 1) and (2, 4). The data set contains the points (1, 1) = (1, 3(1)) and (2, 4) = (2, 3(4)).

The data set seems to represent a vertical stretch of the quadratic parent function by a factor of 3.

1-9 Example 2 Identifying Parent Functions to Model Data Sets

Graph the data from the table.

Describe the parent function and the transformation that best approximates the data set.

x –4 –2 0 2 4

y –12 –6 0 6 12

The graph of the data points resembles the shape of the linear parent function ƒ(x) = x.

The linear parent function passes through the points (2, 2) and (4, 4). The data set contains the points (2, 2) = (2, 3(2)) and (4, 4) = (4, 3(4)).

The data set seems to represent a vertical stretch of the linear function by a factor of 3.

1-9

The cost of playing an online video game depends on the number of months for which the online service is used.

Graph the relationship from number of months to cost, and identify which parent function best describes the data.

Then use the graph to estimate the cost of 5 months of online service.

The linear graph indicates that the cost for 5 months of online service is $72.

Step 1 Graph the relation.

Graph the points given in the table. Draw a smooth line through them to help you see the shape.

Step 2 Identify the parent function.The graph of the data set resembles the shape of a linear parent function ƒ(x) = x.

Step 3 Estimate the cost for 5 months of online service.

1-9 Example 3 Application

HW pg. 70

• 1.9– 2-7 (Sketch Graph), 8, 11-14 (Sketch Graph), 51-55– Ch: 10, 16, 28, 29-39

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chapter 1 - foundations for functions algebra ii

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