Algebra unit 4.2
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Transcript of Algebra unit 4.2
UNIT 4.2 PATTERNS AND LINEARUNIT 4.2 PATTERNS AND LINEAR
FUNCTIONSFUNCTIONS
Warm UpEvaluate each expression for a = 2, b = –3, and c = 8. 1. a + 3c
2. ab – c
3.
12 c + b
4. 4c – b
5. ba + c
26
–14
1
35
17
Identify independent and dependent variables.
Write an equation in function notation and evaluate a function for given input values.
Objectives
independent variabledependent variablefunction rule function notation
Vocabulary
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 and
Example 1 ContinuedStep 2 Determine which relationship works for the other x- and y- values.
10 – 4 ≠ 2 and
15 – 4 ≠ 3 and
20 – 4 ≠ 4 and
The value of y is one-fifth, , of x.
Step 3 Write an equation.or The value of y is one-fifth of x.
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 and 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 value of y is 3 times x.
Step 3 Write an equation.
The value of y is 3 times x.
The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output).
The input of a function is the independent variable. The output of a function is the dependent variable. The value of the dependent variable depends on, or is a function of, the value of the independent variable.
Example 2A: Identifying Independent and Dependent Variables
Identify the independent and dependent variablesin the situation.
A painter must measure a room before deciding how much paint to buy.
The amount of paint depends on the measurement of a room.
Dependent: amount of paintIndependent: measurement of the room
Identify the independent and dependent variablesin the situation.
The height of a candle decrease d centimeters for every hour it burns.
Dependent: height of candle Independent: time
The height of a candle depends on the number of hours it burns.
Example 2B: Identifying Independent and Dependent Variables
A veterinarian must weight an animal before determining the amount of medication.
The amount of medication depends on the weight of an animal.
Dependent: amount of medicationIndependent: weight of animal
Identify the independent and dependent variablesin the situation.
Example 2C: Identifying Independent and Dependent Variables
Helpful Hint
There are several different ways to describe the variables of a function.
IndependentVariable
DependentVariable
x-values y-values
Domain Range
Input Output
x f(x)
Check It Out! Example 2a
A company charges $10 per hour to rent a jackhammer.
Identify the independent and dependent variable in the situation.
The cost to rent a jackhammer depends on the length of time it is rented.
Dependent variable: costIndependent variable: time
Identify the independent and dependent variable in the situation.
Check It Out! Example 2b
Camryn buys p pounds of apples at $0.99 per pound.
The cost of apples depends on the number of pounds bought.
Dependent variable: costIndependent variable: pounds
An algebraic expression that defines a function is a function rule.
If x is the independent variable and y is the dependent variable, then function notation for y is f(x), read “f of x,” where f names the function. When an equation in two variables describes a function, you can use function notation to write it.
The dependent variable is a function of the independent variable.
y is a function of x.
y = f (x)
y = f(x)
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.
Example 3A: Writing Functions
The amount a math tutor charges depends on number of hours.
Dependent: chargesIndependent: hours
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) = 40m + 100.
Example 3B: Writing FunctionsIdentify the independent and dependent variables. Write a rule in function notation for the situation.
The total cost depends on the number of months, plus $100.
Dependent: total costIndependent: number of months
Let m represent the number of months
Check It Out! Example 3aIdentify the independent and dependent variables. Write a rule in function notation for the situation.
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.
Check It Out! Example 3bIdentify 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) = 29.99x + 6.
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 as an input-output machine.
input
10
x
functionf(x)=5x
output
5x
6
30
2
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
Example 4B: Evaluating Functions
Evaluate the function for the given input values.
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
Example 4C: Evaluating Functions
Evaluate the function for the given input values.
For , find h(r) when r = 600
and when r = –12.
= 202 = –2
Check It Out! Example 4a
Evaluate the function for the given input values.
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
Check It Out! Example 4b
Evaluate each function for the given input values.
For g(t) = , find g(t) when t = –24 and
when t = 400.
= –5 = 101
When a function describes a real-world situation, every real number is not always reasonable for the domain and range. For example, a number representing the length of an object cannot be negative, and only whole numbers can represent a number of people.
Example 5: Finding the Reasonable Range and Domain of a Function
Write a function to describe the situation. Find a reasonable domain and range of the function.
Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each.
Money spent is $15.00 for each DVD.f(x) = $15.00 • x
If Joe buys x DVDs, he will spend f(x) = 15x dollars.
Joe only has enough money to purchase 1, 2, or 3 DVDs. A reasonable domain is {1, 2, 3}.
Example 5 Continued
Substitute the domain values into the function rule to find the range values.
x 1 2 3
f(x) 15(1) = 15 15(2) = 30 15(3) = 45
A reasonable range for this situation is {$15, $30, $45}.
Check It Out! Example 5
The settings on a space heater are the whole numbers from 0 to 3. The total of watts used for each setting is 500 times the setting number. Write a function rule to describe the number of watts used for each setting. Find a reasonable domain and range for the function.
Number of watts used
is 500 times the setting #.watts
f(x) = 500 • x
For each setting, the number of watts is f(x) = 500x watts.
x
f(x)
0 1 2 3
500(0) = 0
500(1) = 500
500(2) = 1,000
500(3) = 1,500
There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}.
Check It Out! Example 5
Substitute these values into the function rule to find the range values.
A reasonable range for this situation is {0, 500, 1,000, 1,500} watts.
Lesson Quiz: Part I
Identify the independent and dependent variables. Write a rule in function notation for each situation.
1. A buffet charges $8.95 per person.independent: number of peopledependent: costf(p) = 8.95p
2. A moving company charges $130 for weekly truck rental plus $1.50 per mile.independent: milesdependent: costf(m) = 130 + 1.50m
Lesson Quiz: Part II
Evaluate each function for the given input values.
4. For f(x) = 6x – 1, find f(x) when x = 3.5 and when x = –5.
f(3.5) = 20f(–5) = –31
3. For g(t) = , find g(t) when t = 20 and
when t = –12.g(20) = 2g(–12) = –6
Lesson Quiz: Part III
Write a function to describe the situation. Find a reasonable domain and range for the function.
5. A theater can be rented for exactly 2, 3, or 4 hours. The cost is a $100 deposit plus $200 per hour.
f(h) = 200h + 100Domain: {2, 3, 4}Range: {$500, $700, $900}
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