Module 2 Lecture Notes

MAC1105

Summer B 2019

2 Linear Functions

2.1 Construct a Linear Function From Points

Definition

An in x is a statement that two algebraic expressions are equal.

Definition

A linear equation is the equation of a straight line written in one variable. A linear equation (in

one variable) can be written in the form where a 6= 0.

Note 1. The only power of the variable in a linear equation is 1. For example, x2 � 4 = 0 is not a

linear equation because the power of the variable x is 2.

Definition

The of a line is used to find the steepness and direction of a line. The

vertical change between two points is called the , and the horizontal change

between two points is called the . So, slope =rise

run. Given two pints on a

EQUATION3 1 2 IS NOT ANEQUATIONy Xt 5 ISAN EQUATION

GX b O

5 2 0 IS A LINEAR EQUATION2 2 1 0 IS NOT A LINEAR EQUATION

SLOPE

RISE

RUN

line, (x1, y1) and (x2, y2), the slope of the line between the two points is given by

m =

where m denotes the slope.

Note 2. On a graph, the slope can be represented as:

Example 1. What is the slope of the line below?

Note 3. It does not matter which point you choose as (x1, y1) and which you choose as (x2, y2).

But, make sure you stay consistent with the order of the y terms and the order of the x terms in

2

Yz yXz X

RUN

RISE

m 16X z X

4 UNITSpp

aat.be.UNITS

f Tf LXz Y z

TH y

the numerator and denominator.

Definition

The - of a line is the point at which the line crosses the y-axis. This

is the point where x = 0.

Definition

The - of a line is the point at which the line crosses the x-axis. This

is the point where y = 0.

Note 4. The slope of a line can be positive, negative, undefined, or 0:

3

Y INTERCEPT

Hqy INTERCEPT

X INTERCEPT

1aX INTERCEPT

POSITIVE SLOPE NEGATIVE SLOPEa

n r

L i

v v

a wewoefweosu.pe swp

CANNOTDIVIDE a a 1 BECAUSEBY 0 ANDOUR OUR RISERUN IS 0 150 so50 RISE THE SLOPERise

O IS RISEv v L DRUN RUN

Example 2. Identify the slope, y-intercept, and x-intercept of the line below.

Slope-Intercept Form

The slope-intercept form of an equation is y = mx+ b, where m is the and

b is the -

Note 5. If we are given an equation that is not in slope-intercept form, we can still always find

the y-intercept by plugging in x = 0 and solving for y.

Example 3. Find the equation of the line containing the two points:

(2,�1), (3, 4)

4

y INTERCEPT 1 SLOPE IS RIE II J r

is fo 1 BECAUSEa 1 RUN

ql o

toTHIS THEPOINT 1.5 NEGATIVEWHERE THELINE CROSSESTHE Y AXIS THE X INTERCEPT IS 1 5,0BECAUSE THIS IS THE POINT

WHERE THE LINE CROSSESTHE X AXIS

SLOPE

Y INTERCEPT

X'I'll Xz Yz

1 FINDTHE SLOPE m 4zI I JXz X

2 WRITE IN THE FORM y mXtb MJ so y 5xtbL3 SUBSTITUE IN A POINTTO SOLVE FOR b I WILL USE 2 1

y 5xtb1 512 tb 1 10 tb y 5 14 11

10 101114 y 5

Example 4. Find the equation of the line containing the two points:

(�2, 3), (6,�5)

Point-Slope Form

Given the slope and one point on a line, we can find the equation of the line using point-slope form:

where m is the and (x1, y1) is the point that we are given.

Definitions

• Two lines are if they have the same slope but have di↵erent y-

intercepts.

• Two lines are if the slope of one line is the negative reciprocal of the

other.

Note 6. To find the negative reciprocal of a number, put the number in fraction form and switch

5

4442 ya Yi

1 FIND THE SLOPE m Y 3 8oz lXz X Z 6

2 WRITE AS y mXtb m so y L 1 x b or y X tb

3 SUBSTITUTE A POINTTOSOLVE FORD I WILL USE C 2,3y X t b3 C 2 tb y Xtb y3 2 1 bZ 2 i

y y M X Xi

5 LOPE

PARALLEL

PERPENDICULAR

the numerator and denominator, then negate the number. For example,

How to Find the Equation of Parallel and Perpendicular Lines:

1. Determine the slope of the given line. I suggest writing the equation in slope-intercept form

first.

2. If you are finding a parallel line, the slope is the same as the original line. If you are finding

a perpendicular line, the slope is the of the

original line.

3. Use the point you are given and the slope you found in step 2 to determine the equation of

the line.

6

1 I C F 55

2 I c 3343

NEGATIVE RECIPROCAL

a

g tr

g L gt

PARALLEL PERPENDICULARLINES LINES

Example 5. Write the equation of a line that is parallel to y = 6x + 1 and passes through the

point (�7, 1).

Example 6. Write the equation of a line that is perpendicular to x+ 3y = 6 and passes through

the point (1, 5).

7

O w

gPARALLEL SAME SLOPE m 6

Enstop f6 THE PARALLEL LINE HAS

SUBSTITUTE I 6C 7 t bTOSOLVEFOR b I 42 t b

42 14243TL y6xt43T

1 WRITE IN SLOPE INTERCEPT FORMXt3y 6 Y Eg Izx X

3,1 6 X Y 2 f xI3 T y f x t2y 6SLOPE INTERCEPT FORMSLOPE IS m Iz2 FIND NEW SLOPE PERPENDICULAR NEGATIVE RECIPROCAL

OLD M Iz NEW

3 USE POINT SLOPE FORM WITH M 3 X y 1,5y y M X X

y J 3 X 1

y 5 3 3J to

y 3X

2.2 Converting Between Linear Forms

The Standard Form of a Line

The standard form of a line is where A,B, and C are integers.

Note 7. When the equation of a line is written in standard form, the x and y terms are on one

side of the equal sign, and the constant term is on the other side of the equal sign.

Example 7. Convert the linear function below from standard form to slope-intercept form:

7x+ 5y = 2.

Example 8. Convert the linear function below from slope-intercept form to standard form:

y = �5

4x� 5

2.

8

Axt By C

SLOPE INTERCEPT FORM IS y mxtb

4 75y 7 2F Fy 7x y

WANT Axt By L NO FRACTIONS

y Ex Etoe t E X

F x t y Iz 4 LMULTIPLY BY LCD TO GET RIDOFFRACTIONS

4 Exty 41 EJx t 4y 21 55 4,1107Lx

2.3 Convert Between a Linear Equation and its Graph

How to Determine the Equation of a Line Given its Graph

To find the equation of a line in slope-intercept form given its graph,

1. Find the slope of the line using the formula for the slope between two points.

2. Use the slope and any point on the graph to find the y-intercept, or find the y-intercept on

the graph.

3. Write the equation in the form y = mx+ b.

Example 9. Write the equation of the line below in slope-intercept form and in standard form.

9

1 FIND SLOPE Xi Y

m 3 24 1 2,42

O L 4 y INTERCEPT IS WHEREmIM 11 0 so

y INTERCEPT ISb

Iyy E

2 USE GRAPHTO FIND Y INTERCEPT OR SUBSTITUTE 4 8 W

TO y E X tb TO SOLVE FOR b8 Iy 4 tb8 5 tb5 5

5T y _x

Example 10. Write the equation of the line below in slope-intercept form and in standard form.

Note 8. The equation of a vertical line is given by: . The equation of a horizontal

line is given by: .

10

1 FIND SLOPE

m I y INTERCEPT IS WHERE0 C 2 2 q Xz Yz 11 0 SO

9 y INTERCEPT ISXi Yi 135,74 2 5

2 USE GRAPHTO FIND y INTERCEPT OR SUBSTITUTE 2,10 w

TO 11 52 13 TO SOLVE FOR b

10 52 tb

10 5 tb

o y IzxC

y c

a

y

2.4 Solve Linear Equations

Note 9. To solve an equation is to find all values of x for which the equation is true. Such values

of x are (or roots, zeros) of the equation.

How to Solve Linear Equations in One Variable The steps below do not need to be followed

in any particular order (as long as you remember PEMDAS)

1. Add, subtract, multiply, or divide an equation by a number or expression. BUT, you MUST

do the same thing to both sides of the equal sign. (Math is fair)

2. Apply the distributive property (you may not always need to do this):

3. Isolate the variable on one side of the equation.

4. Solve for the variable by dividing or multiplying a constant.

Example 11. Solve the following equation:

�15(2x+ 10) = �11(�6x+ 14)

11

SOLUTIONS

a btc _ab tac

nd AA2DISTRIBUTIVE

15 2x 1 151 0 111 6 L I1 14 PROPERTY

30 1 150 66 1 1 15430 150 66 15430X t3O GET X ON

ONESIDE150 96 154154 1154

446

EYE

Example 12. Solve the following equation:

�2(9x� 4) = �15(12x+ 10)

Definition

• A is the quotient (or ratio) of two polyno-

mials.

• A is an equation that contains at least one

rational expression.

Note 10. Recall that a rational number is the ratio of two integers. For example,3

2and

5

6are

rational numbers.

Note 11. To solve a linear equation with fractions, multiply both sides by the

to clear the fraction.

12

214 31 231 4 1 15 12 7 C 15 IO

18 8 180 150118011 180162 1 8 150

88162X 158162 162

x II fITRATIONAL EXPRESSION

RATIONAL EQUATION

LEASTCOMMON DENOMINATOR

Note 12. If you are given a rational equation in the form of a proportion:

a

b=

c

d

then you can solve the equation by cross multiplication:

Example 13. Solve the equation:

x� 3

7=

4x+ 12

7

13

aD be

USECROSS MULTIPLICATION

7 x 3 7 4 12ee ee7 21 28 844 721 21 8484 84

105 212 I 21

1 5

Example 14. Solve the equation:

�4x� 5

3� �5x+ 6

2=

3x+ 8

5

14

1 FIND THE LCD LCD IS 3 2.5 302 MULPIPLY BY THE LCD 30 30 4

352 51262 3 00

6Eof 4x y 5

541 30 382

lol 4 5 15 5 16 613 8lol 4 71 14 5 tf 1531 5 31 15 6 613 3 61840 501 75 40 18 4 0 WE35 140 18 48 LIKETERMS ie

10x 18X 40 70 35 150 40 14017 140 48

140 1

14017188II IX188T

Example 15. Solve the equation:

8x� 7

4� �7x+ 7

6=

�7x+ 3

3

15

1 FIND THE LEAST COMMON DENOMINATOR LCD IS 12

2 MULTIPLY BY LCD 12

12 847Htt 7xzt3

Px 8x KyHft If 7x318 7 2 7 17 41 7 3

ee ee ee24 21 14 14 28 12 LONG TERNS ie 24 14 38x

i38 35 28 1221 14 352811 28

66 35 1235 1

35664766 66

4F