Warm Up State whether each word or phrase represents an amount that is increasing, decreasing, or constant. 1. stays the same 2. rises 3. drops 4. slows down

Objectives Vocabulary Match simple graphs with situations. Graph a relationship. Vocabulary continuous graph discrete graph

Objectives Vocabulary Identify functions. Find the domain and range of relations and functions. Vocabulary relation domain function range

Example 1: Relating Graphs to Situations 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. Step 1 Read the graphs from left to right to show time passing.

Check It Out! Ex 2: 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. Step 1 Read the graphs from left to right to show time passing .

As seen in Example 1, some graphs are connected lines or curves called continuous graphs. Some graphs are only distinct points. They are called discrete graphs The graph on theme park attendance is an example of a discrete graph. It consists of distinct points because each year is distinct and people are counted in whole numbers only. The values between whole numbers are not included, since they have no meaning for the situation.

Example 2A: Sketching Graphs for Situations Sketch a graph for the situation. Tell whether the graph is continuous or discrete. A truck driver enters a street, drives at a constant speed, stops at a light, and then continues. As time passes during the trip (moving left to right along the x-axis) the truck's speed (y-axis) does the following: Speed Time y x • initially increases • remains constant • decreases to a stop • increases The graph is continuous.

Example 2B: Sketching Graphs for Situations Sketch a graph for the situation. Tell whether the graph is continuous or discrete. A small bookstore sold between 5 and 8 books each day for 7 days. The number of books sold (y-axis) varies for each day (x-axis). Since the bookstore accounts for the number of books sold at the end of each day, the graph is 7 distinct points. The graph is discrete.

The graph is continuous. Check It Out! Example 2b Sketch a graph for the situation. Tell whether the graph is continuous or discrete. Henry begins to drain a water tank by opening a valve. Then he opens another valve. Then he closes the first valve. He leaves the second valve open until the tank is empty. As time passes while draining the tank (moving left to right along the x-axis) the water level (y-axis) does the following: Water tank Water Level Time • initially declines • decline more rapidly • and then the decline slows down. The graph is continuous.

Both graphs show a relationship about a child going down a slide Both graphs show a relationship about a child going down a slide. Graph A represents the child’s distance from the ground related to time. Graph B represents the child’s Speed related to time.

In Lesson 4-1 you saw relationships represented by graphs In Lesson 4-1 you saw relationships represented by graphs. Relationships can also be represented by a set of ordered pairs called a relation. In the scoring systems of some track meets, for first place you get 5 points, for second place you get 3 points, for third place you get 2 points, and for fourth place you get 1 point. This scoring system is a relation, so it can be shown by ordered pairs. {(1, 5), (2, 3), (3, 2), (4, 1)}. You can also show relations in other ways, such as tables, graphs, or mapping diagrams.

Example 1: Showing Multiple Representations of Relations Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. x y Table Write all x-values under “x” and all y-values under “y”. 2 4 6 3 7 8

Example 1 Continued Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. Graph Use the x- and y-values to plot the ordered pairs.

Example 1 Continued Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. Mapping Diagram x y Write all x-values under “x” and all y-values under “y”. Draw an arrow from each x-value to its corresponding y-value. 2 6 4 3 8 7

Check It Out! Example 1 Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram. Table x y Write all x-values under “x” and all y-values under “y”. 1 3 2 4 3 5

Check It Out! Example 1 Continued Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram. Graph Use the x- and y-values to plot the ordered pairs.

Check It Out! Example 1 Continued Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram. Mapping Diagram x y 1 3 Write all x-values under “x” and all y-values under “y”. Draw an arrow from each x-value to its corresponding y-value. 2 4 3 5

The domain of a relation is the set of first coordinates (or x-values) of the ordered pairs. The range of a relation is the set of second coordinates (or y-values) of the ordered pairs. The domain of the track meet scoring system is {1, 2, 3, 4}. The range is {5, 3, 2, 1}.

Example 2: Finding the Domain and Range of a Relation Ex 1: Give the domain and range of the relation. The domain value is all x-values from 1 through 5, inclusive. The range value is all y-values from 3 through 4, inclusive. Domain: 1 ≤ x ≤ 5 Range: 3 ≤ y ≤ 4

Check It Out! Example 2a Ex 2: Give the domain and range of the relation. 1 2 6 5 The domain values are all x-values 1, 2, 5 and 6. –4 –1 The range values are y-values 0, –1 and –4. Domain: {6, 5, 2, 1} Range: {–4, –1, 0}

x y Check It Out! Example 2b Ex 3: Give the domain and range of the relation. x y 1 4 8 The domain values are all x-values 1, 4, and 8. The range values are y-values 1 and 4. Domain: {1, 4, 8} Range: {1, 4}

A function is a special type of relation that pairs each domain value with exactly one range value.

Example 3A: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain. {(3, –2), (5, –1), (4, 0), (3, 1)} Even though 3 is in the domain twice, it is written only once when you are giving the domain. D: {3, 5, 4} R: {–2, –1, 0, 1} The relation is not a function. Each domain value does not have exactly one range value. The domain value 3 is paired with the range values –2 and 1.

Example 3B: Identifying Functions Ex 1: Give the domain and range of the relation. Tell whether the relation is a function. Explain. {(-4, 2), (-8, 2), (4, 1), (5, 1)} –4 Use the arrows to determine which domain values correspond to each range value. 2 –8 1 4 5 D: {–4, –8, 4, 5} R: {2, 1} This relation is a function. Each domain value is paired with exactly one range value.

Example 3C: Identifying Functions Ex 2: Give the domain and range of the relation. Tell whether the relation is a function. Explain. Draw in lines to see the domain and range values Range Domain D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1 The relation is not a function. Nearly all domain values have more than one range value.

Check It Out! Example 3 Give the domain and range of each relation. Tell whether the relation is a function and explain. Ex 3: {(8, 2), (–4, 1), (–6, 2),(1, 9)} Ex 4: D: {–6, –4, 1, 8} R: {1, 2, 9} D: {2, 3, 4} R: {–5, –4, –3} The relation is a function. Each domain value is paired with exactly one range value. The relation is not a function. The domain value 2 is paired with both –5 and –4.

Lesson Quiz: Part I 1. Express the relation {(–2, 5), (–1, 4), (1, 3), (2, 4)} as a table, as a graph, and as a mapping diagram.

Lesson Quiz: Part II 2. Give the domain and range of the relation. D: –3 ≤ x ≤ 2: R: –2 ≤ y ≤ 4

Lesson Quiz: Part III 3. Give the domain and range of the relation. Tell whether the relation is a function. Explain. D: {5, 10, 15}; R: {2, 4, 6, 8}; The relation is not a function since 5 is paired with 2 and 4.