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Functions & Relations

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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 constant increasing decreasing decreasing

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**Warm Up Generate ordered pairs for the function**

y = x + 3 for x = –2, –1, 0, 1, and 2. Graph the ordered pairs. (–2, 1) (–1, 2) (0, 3) (1, 4) (2, 5)

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**Objectives Match simple graphs with situations. Graph a relationship.**

Identify functions. Find the domain and range of relations and functions.

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**Vocabulary continuous graph discrete graph relation domain range**

function

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**Graphs can be used to illustrate many different situations**

Graphs can be used to illustrate many different situations. For example, trends shown on a cardiograph can help a doctor see how a patient’s heart is functioning. To relate a graph to a given situation, use key words in the description.

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Example 1 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 .

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**Step 2 List key words in order and decide which graph shows them.**

Example 1 Continued Step 2 List key words in order and decide which graph shows them. Key Words Segment Description Graphs… Increased steadily Remained constant Increased slightly before dropping sharply Slanting upward Graph C Graphs A, B, and C Horizontal Slanting upward and then steeply downward Graphs B and C Step 3 Pick the graph that shows all the key phrases in order. The correct graph is graph C.

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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.

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**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.

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**When sketching or interpreting a graph, pay close attention to the labels on each axis.**

Helpful Hint

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**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.

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Try This! Sketch a graph for the situation. Tell whether the graph is continuous or discrete. 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. She improves each week. Each week (x-axis) her typing speed is measured. She gets a separate score (y-axis) for each test. Since each score is separate, the graph consists of distinct units. The graph is discrete.

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**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.

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**You have seen relationships represented by graphs**

You have seen 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.

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**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

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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.

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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

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Try This! 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

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Try This! 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.

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Try This! 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

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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}.

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**Finding the Domain and Range of a Relation**

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

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Try This! 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}

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**x y Try This! Give the domain and range of the relation. 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}

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**A function is a special type of relation that pairs each domain value with EXACTLY ONE range value.**

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**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.

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**Identifying Functions**

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

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**D: {–6, –4, 1, 8} R: {1, 2, 9} D: {2, 3, 4} R: {–5, –4, –3} Try This!**

Give the domain and range of each relation. Tell whether the relation is a function and explain. a. {(8, 2), (–4, 1), (–6, 2),(1, 9)} b. 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.

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**Identifying Functions**

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.

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The "Vertical Line Test" Looking at this function stuff graphically, what if we had the relation {(2, 3), (2, –2)}? We already know that this is not a function, since x = 2 goes to both y = 3 and y = –2. If we graph this relation, it looks like: Notice that you can draw a vertical line through the two points, like this: Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function. Here are a couple examples:

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**The "Vertical Line Test“ continued**

This one is a function. There is no vertical line that will cross this graph twice. This one is not a function. Any number of vertical lines will intersect this oval twice. For instance, the y-axis intersects twice.

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