4-3 writing and graphing functions warm up warm up lesson presentation lesson presentation...

4-3 Writing and Graphing Functions

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4-3 Writing and Graphing FunctionsWarm UpEvaluate each expression for a = 2, b = –3, and c = 8. 1. a + 3c2. ab – c3.

12

c + b

4. 4c – b5. ba + c

26

–141

35

17

6. 2x + y = 3

Solve each equation for y.

7. –x + 3y = –6

8. 4x – 2y = 8

y = –2x + 3

y = 2x – 4


16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. Also covered: 18.0

California Standards


dependent variableindependent variablefunction notation

Vocabulary


Suppose Tasha baby-sits and charges $5 per hour.

Time Worked (h) x 1 2 3 4

Amount Earned ($) y 5 10 15 20

The amount of money Tasha earns is $5 times the number of hours she works. You can write an equation using two variables to show this relationship.

Amount earned is $5 times the number of hours worked.

y =5 x

4-3 Writing and Graphing Functions Additional Example 1: Using a Table to Write an

Equation

Determine a relationship between the x- and y-values. Write an equation.

x

y

5 10 15 20

1 2 3 4

Step 1 List possible relationships between the first x and y-values.

5 – 4 = 1 or

4-3 Writing and Graphing FunctionsAdditional Example 1 Continued

Step 2 Determine which relationship works for the other x- and y- values.

10 – 4 2

15 – 4 3

20 – 4 4

The value of y is one-fifth, , of x.

Step 3 Write an equation.

or The value of y is one-fifth of x.

The second relationship works.


Check It Out! Example 1

Determine a relationship between the x- and y-values. Write an equation.

{(1, 3), (2, 6), (3, 9), (4, 12)}

x

y

1 2 3 4

3 6 9 12

Step 1 List possible relationships between the first x- and y-values.

1 3 = 3 or 1 + 2 = 3


y = 3x

Check It Out! Example 1 Continued

Step 2 Determine which relationship works for the other x- and y- values.

2 • 3 = 63 • 3 = 94 • 3 = 12

2 + 2 6 3 + 2 9 4 + 2 12

The first relationship works. The value of y is 3 times x.

Step 3 Write an equation.

The value of y is 3 times x.


When an equation has two variables, its solutions will be all ordered pairs (x, y) that makes the equation true. Since the solutions are ordered pairs, it is possible to represent them on a graph. When you represent all solutions of an equation on a graph, you are graphing the equation.

Since the solutions of an equation that has two variables are a set of ordered pairs, they are a relation. One way to tell if this relation is a function is to graph the equation use the vertical-line test.

4-3 Writing and Graphing Functions Additional Example 2A: Graphing Functions

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

–3x + 2 = y Step 1 Choose several values of x and generate ordered pairs.

Step 2 Plot enough points to see a pattern.

–3(–1) + 2 = 5

–3(0) + 2 = 2

–3(1) + 2 =–1

–1

0

1

–3x + 2 = y x (x, y)

(–1, 5)

(0, 2)

(1, –1)


Additional Example 2A Continued

Step 3 The points appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.Step 4 Use the vertical line test on the graph.

No vertical line will intersect the graph more than once. The equation –3x + 2 = y represents a function.


When choosing values of x, be sure to choose both positive and negative values.

Helpful Hint

4-3 Writing and Graphing Functions Additional Example 2B: Graphing Functions


y = |x| + 2 Step 1 Choose several values of x and generate ordered pairs.

1 + 2 = 3

0 + 2 = 2

1 + 2 = 3

–1

0

1

|x| + 2 = y x (x, y)

(–1, 3)

(0, 2)

(1, 3)


4-3 Writing and Graphing Functions Additional Example 2B Continued

Step 3 The points appear to form a V-shaped graph. Draw two rays from (0, 2) to show all the ordered pairs that satisfy the function. Draw arrowheads on the end of each ray.

Step 4 Use the vertical line test on the graph.

No vertical line will intersect the graph more than once. The equation y = |x| + 2 represents a function.

4-3 Writing and Graphing FunctionsCheck It Out! Example 2a


y = 3x – 2

Step 1 Choose several values of x and generate ordered pairs.

3(–1) – 2 = –5–1

0

1

3x – 2 = y x (x, y)

(–1, –5)

(0, –2)

(1, 1)

3(0) – 2 = –2

3(1) – 2 = 1



Check It Out! Example 2a ContinuedStep 3 The points appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.Step 4 Use the vertical line test on the graph.

No vertical line will intersect the graph more than once. The equation y = 3x – 2 represents a function.

4-3 Writing and Graphing FunctionsCheck It Out! Example 2b


y = |x – 1|

Step 1 Choose several values of x and generate ordered pairs.

(2, 1)

(–1, 2)

(0, 1)

(1, 0)

(x, y) y = |x – 1| x

2 = |–1 – 1|

1 = |0 – 1|

0 = |1 – 1|

1 = |2 – 1|

–1

0

1

2


4-3 Writing and Graphing FunctionsCheck It Out! Example 2b Continued

Step 3 The points appear to form a V-shaped graph. Draw two rays from (1, 0) to show all the ordered pairs that satisfy the function. Draw arrowheads on the end of each ray.

Step 4 Use the vertical line test on the graph.

No vertical line will intersect the graph more than once. The equation y = |x – 1| represents a function.

4-3 Writing and Graphing FunctionsLooking at the graph of a function can help you determine its domain and range.

All x-values appear somewhere on the graph.

All y-values appear somewhere on the graph.

For y = 5x the domain is all real numbers and the range is all real numbers.

y =5x


Only nonnegative y-values appear on the graph.

Looking at the graph of a function can help you determine its domain and range.

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


In a function, one variable (usually denoted by x) is the independent variable and the other variable (usually y) is the dependent variable. The value of the dependent variable depends on, or is a function of, the value of the independent variable. For Tasha, who earns $5 per hour, the amount she earns depends on, or is a function of, the amount of time she works.

4-3 Writing and Graphing FunctionsWhen an equation represents a function, you can write the equation using functional notation. If x is independent and y is dependent, the function notation for y is f(x), read “f of x,” where f names the function.

The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)Tasha’s earnings, y = 5x, can be rewritten in function notation by substituting f(x) for y—f(x) = 5x. Note that functional notation always defines the dependent variable in terms of the independent variable.


Identify the independent and dependent variables. Write a rule in function notation for the situation.

A math tutor charges $35 per hour.

The function for the amount a math tutor charges is f(h) = 35h.

Additional Example 3A: Writing Functions

The amount a math tutor charges depends on number of hours.

Independent: timeDependent: cost

Let h represent the number of hours of tutoring.


A fitness center charges a $100 initiation fee plus $40 per month.

The function for the amount the fitness center charges is f(m) = 100 + 40m.

Additional Example 3B: Writing Functions


The total cost depends on the number of months, plus $100.

Dependent: total costIndependent: number of months

Let m represent the number of months.

4-3 Writing and Graphing FunctionsCheck It Out! Example 3a


A tutor’s fee for music lessons is $28 per hour for private lessons.

The function for cost of music lessons is f(x) = 28x.

The total cost depends on how many hours of lessons that are given.

Dependent: total costIndependent: lessons given

Let x represent the number of lessons given.


Check It Out! Example 3b


Steven buys lettuce that costs $1.69/lb.

The function for cost of the lettuce is f(x) = 1.69x.

The total cost depends on how many pounds of lettuce that Steven buys.

Dependent: total costIndependent: pounds

Let x represent the number of pounds Steven bought.

4-3 Writing and Graphing FunctionsCheck It Out! Example 3c

Identify the independent and dependent variables. Write a rule in function notation for the situation. An amusement park charges a $6.00 parking fee plus $29.99 per person.

The function for the total park cost is f(x) = 6 + 29.99x.

The total cost depends on the number of persons in the car, plus $6.

Dependent: total costIndependent: number of persons in the car

Let x represent the number of persons in the car.


You can think of a function rule as an input-output machine. For Tasha’s earnings, f(x) = 5x, if you input a value x, the output is 5x.

If Tasha wanted to know how much money she would earn by working 6 hours, she would input 6 for x and find the output. This is called evaluating the function.


Additional Example 4A: Evaluating Functions

Evaluate the function for the given input values.

For f(x) = 3x + 2, find f(x) when x = 7 and when x = –4.

= 21 + 2

f(7) = 3(7) + 2 Substitute 7 for x.

f(x) = 3(x) + 2

= 23

Simplify.

f(x) = 3(x) + 2

f(–4) = 3(–4) + 2 Substitute –4 for x.Simplify.= –12 + 2

= –10


Additional Example 4B: Evaluating Functions


For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2.

g(t) = 1.5t – 5 g(t) = 1.5t – 5

g(6) = 1.5(6) – 5

= 9 – 5

= 4

g(–2) = 1.5(–2) – 5

= –3 – 5

= –8


Additional Example 4C: Evaluating Functions


For , find h(r) when r = 600

and when r = –12.

= 202 = –2


Functions can be named with any letter; f, g, and h are the most common. You read f(6) as “f of 6,” and g(2) as “g of 2.”

Reading Math


Check It Out! Example 4


For h(c) = 2c – 1, find h(c) when c = 1 and when c = –3.

h(c) = 2c – 1

h(1) = 2(1) – 1

= 2 – 1

= 1

h(c) = 2c – 1

h(–3) = 2(–3) – 1

= –6 – 1

= –7


Lesson Quiz: Part I

1. Graph y = |x + 3|.

4-3 Writing and Graphing FunctionsLesson Quiz: Part Il

Identify the independent and dependent variables. Write a rule in function notation for each situation.

2. A buffet charges $8.95 per person.independent: number of peopledependent: costf(p) = 8.95p

3. A moving company charges $130 for weekly truck rental plus $1.50 per mile.independent: milesdependent: costf(m) = 130 + 1.50m


Lesson Quiz: Part III

Evaluate each function for the given input values.

4. For g(t) = find g(t) when t = 20 and when t = –12. g(20) = 2

g(–12) = –6

5. For f(x) = 6x – 1, find f(x) when x = 3.5 and when x = –5. f(3.5) = 20

f(–5) = –31

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