Algebra 1Topic 4:

Relations & Functions

Table of Contents1. Using Graphs to Relate Quantities2. Patterns and Linear Functions3. Nonlinear Functions4. Graphing a Function Rule5. Key Features of Graphs6. Formalizing Relations & Functions

Using Graphs to Relate

Quantities

Which graph could show a car sitting at a stoplight? How do you know?

Miami Traffic

Each day several leaves fall from a tree. One day a gust of wind blows off many leaves. Eventually, there are no more leaves on the tree. Choose the graph that best represents the situation.

Connecting Graphs to Stories

The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation.

Let’s Practice…

Marcus and Janine made the table shown below to represent the difference between their ages during different years.

Which graph matches the information in the table?

Let’s Practice…

Let’s Practice…The table shows the total number of customers at a car wash after 1, 2, 3, and 4 days of its grand opening. Which graph could represent the data shown in the table?

This graph shows someone taking a walk in the neighborhood. Describe what it shows by labeling each part.

Let’s Practice…

Draw Your Own Graph…A pelican flies above the water searching for fish. Sketch a graph of its altitude from takeoff from shore to diving to the water to catch a fish.

Patterns and Linear Functions

VocabIndependent Variable – A variable that doesn’t depend on any other value.– Generally, x is used for the independent

variable.Dependent Variable – A variable whose value depends on some other value.– Generally, y is used for the dependent

variable.

Identify the Independent and Dependent Variables…

1. A painter must measure a room before deciding how much paint to buy.

2. The height of a candle decrease d centimeters for every hour it burns.

3. A veterinarian must weight an animal before determining the amount of medication.

VocabInput – Values of the independent variable.Output – Values of the dependent variable.

Independent Variable Dependent Variable

Vocab Function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value.

Output variable

Input variable

Function Rule

Representing Linear Relationships

Step 1: Make a table.

Step 2: Look for a pattern in the table.

Step 3: Create an equation to represent the pattern.

Step 4: Use the table to make a graph.

A DVD buyers club charges a $20 membership fee and $15 per DVD purchased. The table below represents this situation.

Number of DVDs purchased x 0 1 2 3 4 5

Total cost ($) y 20 35 50 65 80 95

What’s the Function?

Let’s Practice…The table shows the amount of

water y in a tank after x minutes of being drained. 1. Is the relationship function?

2. Describe the relationship using words.

3. Write an equation for the relationship.

Nonlinear Functions

Nonlinear FunctionNonlinear Function: a function whose graph is not a line or part of a line.

Is the graph is linear or nonlinear?

A. B.

Let’s Practice…

Is the graph is linear or nonlinear?

C. D.

Let’s Practice…

What’s Special About Nonlinear Functions?!?!

• Linear functions have a constant relationship between x and y.

• HOWEVER, nonlinear functions, the differences between x-values and the corresponding y-values are not the same.

• Yet, there still is a pattern….

What’s the pattern??

Tell whether the function in the table has a linear or nonlinear relationship.

Let’s Practice…

What’s the pattern??

Tell whether the function in the table has a linear or nonlinear relationship.

Let’s Practice…

Input Output

1 2

2 5

3 11

Writing a Rule to Describe a Nonlinear Function

Step 1: Make a table to organize the x and y values.

Step 2: Look for a pattern in the y-values and identify a rule that produces the given y-value when you substitute the x-value.

Step 3: Write and verify the function rule.

Let’s Practice…Write a function rule for the nonlinear function:

x y1 2

2 4

3 8

4 16

5 32

Let’s Practice…The ordered pairs (1, 1), (2, 4), (3, 9), (4, 16), and

(5, 25) represent a function. What is a rule that represents this function?

Graphing a Function Rule

VocabDiscrete: A discrete function is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4,...

Continuous: A continuous random variable is one which takes an infinite number of possible values.• Continuous random variables are usually measurements.

Car Approaching Traffic Light

Henry begins to drain a water tank by opening a valve.

Water tank

Wat

er L

evel

Time

Discrete or Continuous? Jamie is taking an 8-week keyboarding class. At the end of each week, she takes a test to find the number of words she can type per minute.

Procedure: Graphing Functions

Graphing a Line1. Pick three x-values. • One should be negative, one should be

zero and one should be positive. 2. Plug in the x-values to the function to

determine y value. 3. Graph each ordered pair. 4. Connect the dots with a line.

Graph y = 2x + 1.

Let’s Practice…

X Y

Graph y = 4x.

Let’s Practice…

X Y

Let’s Practice…Graph 15x + 3y = 9.

X Y

The relationship between human years and dog years is given by the function y = 7x, where x is the number of human years. Graph this function.

Dog Years…

X Y

At a salon, Sue can rent a station for $10.00 per day plus $3.00 per manicure. The amount she would pay each day is given by f(x) = 3x + 10, where x is the number of manicures. Graph this function.

Let’s Practice…

X Y

Graphing Nonlinear Functions

1. Pick 5 x-values. • Negatives, positives and zero.

2. Plug in the x-values to the function to determine y-value.

3. Graph each ordered pair. 4. Connect the dots with a line.

Let’s Practice…Graph the function f(x) = |x| + 2.

X Y

Graph the function y = |x – 1|.

Let’s Practice…

X Y

Graph the function y = x2.

Let’s Practice…

X Y

Graph the function y = x2 + 1.

Let’s Practice…

X Y

Key Features of Graphs

Key Features of GraphIntercepts X- Intercepts

Y- InterceptsRelative Extremes Relative Maximum

Relative MinimumSymmetry y-axis

x-axisy=x

Slope PositiveNegative

Form IncreasingDecreasing

InterceptsX-intercept: Where the graph crosses the x axis. The y-value is zero.Y-intercept: Where the graph cross the y axis. The x-value is zero.

Let’s Practice…What are the x-intercept and y-intercept?

Relative ExtremesRelative Maximum: highest point on the graph around it (looks like a hill)Relative Minimum: lowest point on the graph around it (looks like a valley)

Let’s Practice…What is a relative maximum of this graph?What is a relative minimum of this graph?

Symmetryx-axis: When flipped over the x-axis, the graph looks the same

E

Symmetryy-axis: When flipped over the y-axis, the graph looks the same

Symmetryy=x: When flipped over the line y=x, the graph looks the same

Let’s Practice…What kind of symmetry does the picture have?

SlopePositive: the graph increases from left to rightNegative: the graph decreases from left to right

Negative

Positive

Let’s Practice…

A B

C

Of lines A, B, and C, which lines are positive?Which lines are negative?

FormIncreasing: The values of y get bigger from left to rightDecreasing: The values of y get smaller from left to right

Let’s Practice…

Name the internals where the function is increasing and decreasing.

Formalizing Relations and

Functions

2

4

6

3

7

8

x yTable

RelationRelation – A relationships that can be represented by a set of ordered pairs.

Example: Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.

2

6

4

3

8

7

x y

Domain & RangeDomain: The set of first coordinates (or x-values) of the ordered pairs of a relation.

Range: The set of second coordinates (or y-values) of the ordered pairs of a relation.

Only nonnegative y-values appear on the graph.

All x-values appear somewhere on the graph.

For y = x2 the domain is all real numbers and the range is y ≥ 0.

y = x2

Example: Finding Domain and Range from a Graph

Give the domain and range of the relation.

Let’s Practice…

Give the domain and range of the relation.

x y

1 1

4 4

8 1

Let’s Practice…

–4

–1

01

2

6

5

FunctionsFunction: a special type of relation that pairs each domain value with exactly one range value.

All functions are relations, but all relations are not functions.

Give the domain and range of the relation. Tell whether the relation is a function. Explain.

1. {(3, –2), (5, –1), (4, 0), (3, 1)}

2. {(-1, 5), (0, 2), (1, 3), (5, 7)}

3. {(3, 9), (4, -6), (0, 9), (-1, -5)}

Is This Relation a Function?

Give the domain and range of the relation. Tell whether the relation is a function. Explain.

1. {(3, –2), (5, –1), (4, 0), (3, 1)}

2. {(-1, 5), (0, 2), (1, 3), (5, 7)}

3. {(3, 9), (4, -6), (0, 9), (-1, -5)}

Is This Relation a Function?

D: {3, 4, 5} R: {-2, -1, 0, 1}

Yes, since each domain value does not repeat.

Yes, since each domain value does not repeat.

No. The 3 connects to both the -2 and 1.

D: {-1, 0, 1, 5} R: {2, 3, 5, 7}

D: {-1, 0, 3, 4} R: {-5, -6, 9}

–4

–8

4

2

1

5

Is This Relation a Function?Give the domain and range of the relation. Tell whether the relation is a function. Explain.

a. {(8, 2), (–4, 1), (–6, 2),(1, 9)}

b.

Let’s Practice…Give the domain and range of the relation. Tell whether the relation is a function. Explain.

Let’s Practice…Give the domain and range of each relation. Tell whether the relation is a function and explain.

Short Cut: Vertical Line Test

Graph each equation. Then tell whether the equation represents a function.

y = 3x – 2

Let’s Practice…

X Y

Graph each equation. Then tell whether the equation represents a function.

y = |x – 1|

Let’s Practice…

X Y

Function Notationf(x) is a fancy way of saying “y”

Evaluating Functions & Finding the Range

• We can evaluate a function and find the range by plugging in!

Example:Find the range: f(x) = 3x + 2 using (1, 2, 3, 4)

Let’s Practice…Find the range:

1. f(x) = 2x -7 (-2, -1, 0, 1 , 2)

2. h(x) = x2; (-5, 0, 3, 4, 7)