Algebra 1

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Algebra 1

• Chapter 4 Section 3

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4-3 Writing and Graphing Functions

• Objectives
• Write an equation in function notation and
  evaluate a function for given input values.
• Graph functions and determine whether an equation
  represents a function.

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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
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4-3 Writing and Graphing Functions
Suppose Tasha baby-sits and charges 5 per hour.


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.
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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.
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Example 1 Continued
Step 2 Determine which relationship works for the
other x- and y- values.
?
?
?
?
?
?
The second relationship works.
Step 3 Write an equation.
The value of y is one-fifth of x.
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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.
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Example 2 Graphing Functions
Graph each equation. Then tell whether the
equation represents a function.
Step 2 Plot enough points to see a pattern.
3x 2 y
Step 1 Choose several values of x and generate
ordered pairs.
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Example 2 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.
?
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Example 3 Graphing Functions
Graph each equation. Then tell whether the
equation represents a function.
Step 2 Plot enough points to see a pattern.
y x 2
Step 1 Choose several values of x and generate
ordered pairs.
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Example 3 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.
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Looking at the graph of a function can help you
determine its domain and range.
y 5x
For y 5x the domain is all real numbers and the
range is all real numbers.
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Looking at the graph of a function can help you
determine its domain and range.
y x2
All x-values appear somewhere on the graph.
For y x2 the domain is all real numbers and the
range is y 0.
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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.
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When 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.
Tashas 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.
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Example 4 Writing Functions
Identify the independent and dependent variables.
Write a rule in function notation for the
situation.
A math tutor charges 35 per hour.
The amount a math tutor charges depends on number
of hours.
Independent time Dependent cost
Let h represent the number of hours of tutoring.
The function for the amount a math tutor charges
is f(h) 35h.
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Example 5 Writing Functions
Identify the independent and dependent variables.
Write a rule in function notation for the
situation.
A fitness center charges a 100 initiation
fee plus 40 per month.
The total cost depends on the number of months,
plus 100.
Dependent total cost Independent number of
months
Let m represent the number of months.
The function for the amount the fitness center
charges is f(m) 100 40m.
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You can think of a function rule as an
input-output machine. For Tashas 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.
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Example 6 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.
f(x) 3(x) 2
f(x) 3(x) 2
Substitute 7 for x.
f(4) 3(4) 2
Substitute 4 for x.
f(7) 3(7) 2
21 2
Simplify.
Simplify.
12 2
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