Algebra I Notes Writing Functions OBJECTIVES -...

Algebra I Notes Writing Functions

Writing Functions_Notes of 20 9/11/2014

OBJECTIVES:

F.BF.A.1 Build a function that models a relationship between two quantities.

Write a function that describes a relationship between two quantities.

F.IF.A. 2. Understand the concept of a function and use function notation.

Use function notation, evaluate functions for inputs in their domains, and interpret

statements that use function notation in terms of context.

F.IF.B.4 Interpret functions that arise in applications in terms of the context.

For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing

key features given a verbal description of the relationship.

F.IF.B.6 Calculate and interpret the average rate of change of a function (presented

symbolically or as a table) over a specified interval. Estimate the rate of change

from a graph.

BIG IDEA:

Relationships among quantities can be represented using tables, graphs, verbal

descriptions equations and inequalities. Symbols are used to represent unknowns and

variables. We can interpret and make critical predictions from functional relationships.

PREREQUISITE SKILLS:

students know how to translate sentences into equations

students know how to present relations and functions

students know how to find function values

students know how to locate & name points on a coordinate plane using ordered pairs

students know how to represent relationships among quantities using graphs

students understand function notation

students understand how to solve and graph equations in one or two variables

VOCABULARY:

independent variable: the variable in a function with a value that is subject to choice

dependent variable: the variable in a relation with a value that depends on the value of

the independent variable (input)

function rule: an algebraic expression that defines a function

function notation: a way to name a function that is defined by an equation. In function

notation, the y in the equation is replaced with f(x)



SKILLS:

identify independent and dependent variables

create, use, translate and make connections among algebraic, tabular, graphical, or verbal

descriptions of linear functions

write the equation of a linear function given two points, a point and the slope, a table of

values or a graphical representation

use properties of functions and attributes of functions and apply functions to problem

situations

solve equations that can be used for prediction or interpretation of applications

translate among the different forms of linear equations including slope-intercept, point-

slope, and standard form

identify parallel, perpendicular, and intersecting lines by slope

REVIEW AND EXAMPLES:

Ex 1: Identify the independent and the dependent variables in the examples below.

In the winter, more electricity is used when the temperature goes down, and less is used

when the temperature rises.

Since the amount of electricity changes depending on the temperature, it is the

dependent variable or output (y). The temperature is the independent variable or

input (x).

The faster Dave walks, the quicker he gets home.

Since the time it takes to get home depends on how fast Dave walks, time is the

dependent variable or output (y). The speed Dave walks is the independent

variable.

Ex 2: Suppose Annie babysits and charges $4.50 per hour. The amount of money that she

earns is $4.50 times the number of hours she works. Write an equation using two

variables to show this relationship.

Annie’s total money earned depends on the number of hours she babysits, so the number

of hours is the independent variable (input) and the total money earned is the dependent

variable (output). Her money earned is a function of how many hours she babysits.

$ $ earned multiply # of hours

earned equals per hour by worked

y = $4.50 × x



Using the equation, Annie could determine how much money she will earn for any

number of hours she works. Let’s determine how much she would earn if she worked 6

hours and 8.5 hrs.

First, let’s rewrite the equation in function notation. Since y ($ earned) is a function of x

(hours worked), we replace y with f(x). The function rule is the math we must do to the

input (x). So, the function rule is $4.50x.

y = $4.50x

f(x) = $4.50x Now, substitute in the given values of x and evaluate the function rule.

f(6) = $4.50(6) f(8.5) = $4.50(8.5)

f(6) = $27 f(8.5) = $38.25

Ex 3: Tim has already sold $40 worth of tickets to the school musical. Tim has 4 tickets left to

sell at $2.50 per ticket. Write a function to describe how much money Tim can collect

from selling tickets.

Tim’s total money collected depends on the number of tickets he sells, so the number of

tickets sold is the independent variable (input) and the total money collected is the

dependent variable (output). His money collected is a function of how many tickets he

sells plus the $40 he has already collected.

$ cost multiply # of tickets $ already

collected equals per ticket by sold plus collected

y = $2.50 × x + $40

f(x) = $2.50 × x + $40

Determine a reasonable domain and range for this function.

Since Tim only has 4 tickets left to sell, he has a limited domain. He can sell 0, 1 , 2 , 3

or 4 whole tickets (he can’t sell pieces of tickets). Therefore the domain is {0, 1, 2, 3, 4}.

To find the range, we input each domain value into the function rule and evaluate.

f(x) = 2.50x + 40 f(x) = 2.50x + 40 f(x) = 2.50x + 40 f(x) = 2.50x + 40 f(x) = 2.50x + 40

f(0) = 2.50(0) + 40 f(1) = 2.50(1) + 40 f(2) = 2.50(2) + 40 f(3) = 2.50(3) + 40 f(4) = 2.50(4) + 40

f(0) = 0 + 40 f(1) = 2.50 + 40 f(2) = 5 + 40 f(3) = 7.50+ 40 f(4) = 10 + 40

f(0) = 40 f(1) = 42.50 f(2) = 45 f(3) = 47.50 f(4) = 50

The range for this situation is {$40, $42.50, $45, $47.50, $50}



Ex 4: Use the given table of values to determine a relationship between x and y and write an

equation for the function. Look for the pattern in the data.

a. b.

The pattern suggests that for every increase The pattern suggests that for every increase of

of 1 in x, y increases by 2. So the slope is 2. 1 in x, y decreases by 3. So, the slope is −3.

To find the y-int. (b) we can work backwards Since we know the value of y when x is 0,

using the pattern of the slope to determine the we already know the y-int. (b). It is −1.

value of y when x is 0. When x is 0, y is −4.

*Now, we can write the function rule: *Now, we can write the function rule:

slope-intercept: y = 2x – 4 slope-intercept: y = −3x – 1

function notation: f(x) = 2x – 4 function notation: f(x) = 3x – 1

What is the greatest whole number the input could be in function (a) to have an output that is

less than 9? f(x) = 2x – 4

9 > 2x – 4

13 > 2x

6.5 > x 6 is the greatest whole number that the input could be.

Slope-intercept form of a linear equation:

When we write a linear equation in function notation, the function rule or right side of the

equation is just like an equation written in slope-intercept form.

y mx b , where m is the slope and b is the y-intercept

If you know the slope of the line and the y-intercept (initial constant amount), you can write the

equation.

Ex 5: Write the equation of a line with a slope of 4 and a y-intercept of −6.

Use slope-intercept form with m = 4 and b = −6.

y mx b 4 6 or 4 6y x y x

function notation f(x) = 4x – 6

If you have the graph of a linear equation you can determine the slope and y-intercept of the

function and write an equation.

x 1 2 3 4 5

f(x) -2 0 2 4 6

x 2 1 0 1 2

g(x) 5 2 1 4 7



Ex 6: Write the equation of the line shown in the graph.

The y-intercept is 3, as seen in the graph.

To find the slope, use the two points shown in the graph.

Starting from the point on the left, the rise is down 2 and

the run is right 4, so the slope is 2 1

4 2

.

Use slope-intercept form with 1

and 32

m b .

slope-intercept: y mx b 1

32

y x

function notation: 1

( ) 32

f x x

Ex 7: Xavier weighed 220 pounds and lost 2 pounds a month for 6 months. Write a linear

equation to model Xavier’s weight (w), over m months. Use the model to find how much

Xavier weighed after 6 months.

The y-intercept represents Xavier’s beginning weight at 0 months: b = 220

The slope (rate of change) represents Xavier’s change in weight each month. (Note:

Because he is losing weight, the slope will be negative.) 2m

slope-intercept: 2 220w m

function notation: ( ) 2 220w m m

To find Xavier’s weight after 6 months, let m = 6. (6) 2 6 220 208 lbsw

Ex 8: A taxi charges a flat fee of $10 plus $1.50 per mile. Write a linear equation that

represents the cost, c, of a taxi ride for m miles.

The y-intercept represents the initial fee (at 0 miles): b = 10

The slope (rate of change) represents the fee per mile: m = 1.50

slope-intercept: 1.5 10c m

function notation: ( ) 1.5 10c m m



Point-Slope Form of a Linear Equation:

1 1y y m x x m = slope, 1 1,x y = point on the line

Teacher Note: This form is used frequently in subsequent courses – be sure to emphasize!

Point-slope is simply the slope formula rewritten.

2 1

2 1

y ym

x x

by cross-multiplying, we get 2 1 2 1m x x y y

Writing an Equation of a Line Given a Point and the Slope

Ex 9: Write an equation of the line that passes through the point 2, 1 and has a slope of 1

2

in slope-intercept form and in function notation.

Use point-slope form with 1 1, 2, 1x y and 1

2m .

1 1y y m x x 1 1

1 2 1 22 2

y x y x

To convert to slope-intercept form, solve the point-slope form equation for y.

1

1 22

y x 1 1

1 1 22 2

y x y x

function notation 1

( ) 22

f x x

Ex 10: Write an equation of the line that passes through the point 3,5 and has a slope of 7 in

point-slope form. Then convert to slope-intercept and finally write it in function

notation.

Use point-slope form with 1 1, 3,5x y and 7m .

point-slope: 1 1y y m x x 5 7 3 5 7( 3)y x y x

y – 5 = 7x + 21

slope-intercept y = 7x + 26

function notation f(x) = 7x + 26



Writing an equation of a line given two points

Step One: Find the slope using the two points in the slope formula.

Step Two: Use the slope and one of the points (either one) in the point-slope equation.

Step Three: Write the equation in the form required for the problem.

Ex 11: Write the equation of the line that passes through the points 4,7 and 1, 5 in slope-

intercept form.

Step One: Find the slope using the two points in the slope formula.

7 5 124

4 1 3m


We will use the point 4,7 .

1 1y y m x x 7 4 4 7 4 4y x y x

Step Three: Write the equation in the form required for the problem.

7 4 16

4 9

y x

y x

Ex 12: Work the problem above again, using 1, 5 as the point in the point-slope equation.

This time, write the equation in function notation.

We will use the point 1, 5 .

1 1y y m x x ( 5) 4 1 5 4 1y x y x

5 4 4

4 9

y x

y x

f(x) = 4x – 9



Writing an Equation of a Line Given the Graph

Ex 13: Write the equation of the line shown in the graph in function notation.

We will follow the same steps in the previous example using the points on

the graph.

Step One: Find the slope using the two points on the graph. Starting at the

point on the left, we would step up 3 and right 2 to get to the other point, so

the slope is 3

2m .


We will use the point 3,0 . 1 1y y m x x 3 3

0 3 32 2

y x y x

Step Three: Write the equation in the form required for the problem. 3 9

2 2y x

3 9

( )2 2

f x x

Slopes of Parallel Lines

Ex 14: Graph the following lines by hand on the same coordinate plane.

3 6y x 2 3 4y x 6 2 12x y

m = 3; b = 6 pt. (−4, 2); m = 3 x-int: 2; y-int: −6

What do you notice about the lines? (They are parallel.)

Find the slope of the lines using the equation. What do you

notice about the slopes? (They are the same.)

Conclusion: Parallel lines have equal slopes.

Ex 15: Write an equation of the line parallel to the line 2 1y x that passes through the point

4, 5 in point-slope form, slope-intercept form and function notation.

Step One: Determine the slope of the given line. 2 1y x is in slope-intercept form with

2m .

Step Two: Determine the slope of the line parallel to the given line. Because parallel lines have

equal slopes, the slope of the line parallel is also 2m .



Step Three: Write the equation of the parallel line in point-slope form using the slope found in

Step Two and the point given in the problem. Convert to slope-intercept and function notation.

point-slope: 1 1y y m x x 5 2 4 5 2( 4)y x y x

slope-intercept: y = mx + b 5 2( 4)y x 5 2 8y x 2 3y x

function notation: ( ) 2 3f x x

Writing Equations of Perpendicular Lines

Perpendicular Lines: two lines that intersect at a right angle.

Perpendicular lines have slopes that are opposite reciprocals.

Ex 16: Write an equation of the line perpendicular to the line 1

7 13

y x that passes through the

point 1,7 in slope-intercept form and function notation.

Step One: Determine the slope of the given line. 1

7 13

y x is in point-slope form with

1

3m .

Step Two: Determine the slope of the line perpendicular to the given line. Because perpendicular

lines have slopes that are opposite reciprocals, the slope of the line perpendicular is 3m .

Step Three: Write the equation of the perpendicular line in point-slope form using the slope found

in Step Two and the point given in the problem.

1 1y y m x x 7 3 1 y x

Step Four: Write the equation in slope-intercept form. 7 3 3 3 10y x y x

Step Five: Write the equation in function notation. 3 10 ( ) 3 10y x f x x

Graph the two lines to show they are perpendicular.

y – 7 = 1

3(x – 1) ; m =

1

3; pt (1, 7)

y = −3x + 10 ; m = −3 ; b = 10



Standard Form of a Linear Equation

Ax By C ; where A, B, and C are real numbers and A and B are not both zero

Ex 17: Write the equation 3

54

y x in standard form with integer coefficients.

Step One: Wipe out fractions to create integer coefficients by multiplying each term in the

equation by 4.

3

4 4 4 5 4 3 204

y x y x

Step Two: Rewrite the equation so that the x and y terms are on one side, and the constant term is

on the other side.

4 3 20

3 4 20

y x

x y

or

4 3 20

20 3 4

3 4 20

y x

x y

x y

Note: There is more than one way to write an equation of a line in standard form!

Writing a Linear Equation in Standard Form Given Two Points

Ex 18: Write an equation of the line that passes through the points 2,4 and 1, 5 in standard form.

Step One: Find the slope. 4 5 9

32 1 3

m

Step Two: Use the slope and one of the points given to write the equation in point-slope form.

4 3 2y x 4 3 2y x

Step Three: Rewrite the equation in standard form.

4 3 9

3 5

y x

x y

Writing a Linear Equation in Standard Form Given the Graph

Ex 19: Write the equation of the line shown in the graph in standard form.

This is a vertical line with an x-intercept of −4, so the equation is 4x .



Ex 20: Giovanni has $20 to spend on apples and grapes at the grocery store. Apples cost $2.00 per pound

and grapes cost $1.25 per pound. Write a linear equation in standard form that models the

different amounts of apples, a, and grapes, g, that Giovanni can buy for $20. Graph the line.

Verbal Model: Price of Apples × Pounds of Apples + Price of Grapes × Pounds of Grapes = Total Cost

Labels: Price of Apples = 2 Pounds of Apples = a Price of Grapes = 1.25 Pounds of Grapes = g

Total Cost = 20

Algebraic Model: 2 1.25 20a g

Graph: Let a be the horizontal axis and g be the vertical axis.

The a-intercept is 10 (let g = 0), and the g-intercept is 16 (let a = 0).

Any point on the line is a possible combination of apples and grapes he

can buy.



ASSESSMENT ITEMS:

1. The table below shows the total amount a repair company charges for five different jobs based

on the number of hours each job takes to complete.

The relationship between the number of hours a job takes to complete and the total

amount the repair company charges continues with the same pattern. What is the least

number of hours the repair company could take to complete a job and charge a total

amount that is greater than $5,000?

A 77 hours

B 114 hours

C 124 hours

D 126 hours

ANS: C

2. The profit for the school play is $4 per ticket minus $280 for the expense to build the set. There

are 300 seats in the theater.

a) Write an equation that represents the profit (p) for n tickets sold.

Solution: p = 4n – 280

b) Describe the domain of the function.

Solution: , 70 300n n n

Read: “n is an element of the set of whole numbers, such that n

is greater than orequal to 70 and less than or equal to 300.”

c) Make an input-output table. (Note: input values may vary)

Solution:

n 70 100 150 200 300

p 0 120 320 520 920



d) Describe the range of the function.

Solution: { , multiples of 4 0 920p p p

Read: “p is an element of the set of multiples of 4, such that p

is greater than or equal to 0 and less than or equal to 920.”

3. It takes 150 hours to resurface 30 miles of road.

a) Write a linear function for resurfacing d miles of road in t hours.

Solution: 1

5d h

b) Create an input-output table for the function.

Solution: tables may vary.

h 0 10 50 100 150

d 0 2 10 20 30

c) Graph the function.

Solution:

d) Find the length of the road done in 500 hours.

Solution:

1

5

1(500)

5

100 mi

d h

d

d



4. Translate the table into words:

A. The output is three more than negative four times the input.

B. The output is negative three times the input.

C. The output is two less than negative three times the input.

D. The output is three less than negative four times the input.

ANS: A

5. Which sentence represents the equation y = 3x – 7, where y is Devon’s age and x is his sister’s

age?

A. Devon is 3 years younger than seven times the age of his sister.

B. Devon is 3 years older than seven times the age of his sister.

C. Devon is 7 years younger than three times the age of his sister.

D. Devon is 7 years older than three times the age of his sister.

ANS: C

6. Identify the independent and dependent variables in each situation:

a. A painter must measure a room before deciding how much paint to buy.

Solution: independent variable is measurement of room

dependent variable is amount of paint

b. The height of a candle decreases for every hour it burns.

Solution: independent variable is hours the candle burns

dependent variable is height of the candle

c. A lawyer charges a set hourly fee for services.

Solution: independent variable is hours of service

dependent variable is total fee charged by lawyer

7. Write an equation in function notation for each situation:

a. To rent a DVD, a customer must pay $3.99 plus $0.99 for every day that it is late.

Solution: f(x) = $0.99x + $3.99

b. Stephen charges $25 for each lawn he mows.

Solution: f(x) = $25x

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

Solution: f(x) = $40x + $100

Input 3 4 5 6 7

Output –9 –13 –17 –21 –25



8. A car can travel 30 miles on a gallon of gas and has a 20-gallon gas tank. Let g be the number

of gallons of gas the car has in its tank.

a. Write a function that gives the distance d in miles that the car travels on g gallons of gas.

Solution: d = 30g

b. What are reasonable values for the domain and range in the situation described?

Solution: domain:{0 < g < 20}

range: {0 < d < 600}

c. How far can the car travel on 12 gallons of gas?

Solution: 360 mi

9. Marsha buys x pens at $0.70 per pen and one pencil for $0.10. Which function gives the total

amount Marsha spends?

A. c(x) = 0.70x + 0.10x

B. c(x) = 0.70x + 1

C. c(x) = (0.70 + 0.10)x

D. c(x) = 0.70x + 0.10 ANS: D

10. A mail-order company charges $5 per order plus $2 per item in the order, up to a maximum of

4 items. What is the reasonable domain and range for the function?

A. domain: {1, 2, 3, 4}; range: {2, 4, 6, 8}

B. domain: {1, 2, 3, 4}; range: {7, 9, 11, 13}

C. domain: {0, 1, 2, 3, 4}; range: {0, 2, 4, 6, 8}

D. domain: {0, 1, 2, 3, 4}; range: {0, 7, 9, 11, 13} ANS: B

11. Write the equation of a line with a slope of 1

3 and a y-intercept of −5 in slope-intercept form

and then in function notation.

Solution: slope-intercept: 1

53

y x ; function notation: 1

( ) 53

f x x



12. Write the equation of the line shown in the graph in slope-intercept form and then in function

notation.

A. B.

Solution: slope-intercept: y = 2x – 5 slope-intercept: y = 4

function notation: f(x) = 2x – 5 function notation: f(x) = 4

y

13. When writing a linear model for an application problem, describe what the slope and y-

intercept represent.

Solution: the slope represents the pattern of the function or change in y for each increase of

1 in x. The y-intercept represents the initial amount when x is 0.

14. What is the equation of the line in slope-intercept form passing through the points in the table?

x −2 0 2 4 6

y −3.5 −3 −2.5 −2 −1.5

A. 1

32

y x

B. 1

42

y x

C. 1

14

y x

D. 1

34

y x ANS: D



15. Use the graph below.

What is the equation of the line in the graph?

A. 2 1y x

B. 2 2y x

C. 2 1y x

D. 2 2y x ANS: D

16. Write the equation of the line parallel to the line 1

83

y x that passes through the point

9,6 in slope-intercept form and function notation.

Solution: 1

3m . Point-slope

16 ( 9)

3y x

Slope-intercept 1 1

6 3 93 3

y x y x

Function notation 1

( ) 93

f x x

17. Describe the relationship of the equations of parallel lines and perpendicular lines.

Solution: parallel lines have the same slope and perpendicular lines have slopes that are

negative (opposite) reciprocals.



18. Write the equation 1

3 ( 4)2

y x in slope-intercept form.

A. 1

12

y x

B. 1

12

y x

C. 1

52

y x

D. 1

72

y x

ANS: C

19. Identify a point-slope equation of the line that passes through the point (–4, 5) and has a slope

of 3

10 .

A. 3

4 510

y x

B. 3

5 410

y x

C. 10

5 43

y x

D. 10

4 53

y x

ANS: B

20. Which is an equation of a line perpendicular to 2 1

5 3y x ?

A. 2

35

y x

B. 5

22

y x

C. 5

32

y x

D. y = 3x + 3

ANS: C



21. Which equation represents the line that contains the point (–3, 14) and is parallel to the line

2 5x y ?

A. 14 2 3y x

B. 14 2 3y x

C. 1

14 32

y x

D. 1

14 32

y x

ANS: A

22. Write an equation in standard form with integer coefficients of the line that passes through the

points 9, 2 and 3,2 .

Solution: Step One: find the slope; 2 ( 2) 4 1

3 9 12 3m

Step Two: use the slope and a point in point-slope form 1

2 ( 9)3

y x

Step Three: convert point-slope to slope-intercept 1

13

y x

Step Four: clear the fractions 1

3 3 1 33

y x 3 1 3y x

Step Five: write in standard form 3 3x y

23. Write an equation in standard form of the horizontal line and the vertical line that pass through

the point 8,9 .

Solution: horizontal line 9y

vertical line 8x

24. What is the y-intercept of 4 3 24x y ?

A. 8

B. 6

C. 6

D. 8

ANS: A



25. Find the x- and y-intercepts of 5 3 15y x .

A. int : 5; int : 3x y

B. int : 5; int : 3x y

C. int : 5; int : 3x y

D. int : 5; int : 3x y

ANS: B

26. The graph shows membership costs at a gym. How much was the initial membership fee?

27. The graph shows membership costs at a gym. What is the cost per month?

Solution:

Find the slope: 205 100

355 2

m

Point-slope 100 35( 2)y x

Slope-intercept 35 30y x

y-int (initial membership fee) = $30

Solution: cost per month is the slope.

Find the slope: 205 100

$35 / month5 2

m

Algebra I Notes Writing Functions OBJECTIVES -...

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