Algebra I Unit 2 Linear RelationshipsChapter 3 Linear Functions

Lesson 3-1 Graphing Linear Equations

Objectives: I can identify linear equations, intercepts, and zeros.

I can graph linear functions.

CCSS: F.IF.4, F.IF.7a, MP.8

Example 1: Identify Linear EquationsDetermine whether each equation is a linear equation. Write the equation in standard form.

a.

b.

c. Guided Practice 1: Identify Linear EquationsDetermine whether each equation is a linear equation. Write the equation in standard form.

a.

b.

c.

Standardized Test Example 2: Find Intercepts from a Graph

Find the x- and y-intercepts of the segment graphed below.

÷linear 5y -1×+2=0+ ×

+ ×- z - z

Ax + By = C0

× -1 5y= 2

Not linear

0 Not linear

• 3 -3

yes linear

y= -3

Axt By :Not linear

# Not linear !

A x-intercept is 0; y-intercept is 6 B x-intercept is -3; y-intercept is 0

C x-intercept is -3; y-intercept is 6 D x-intercept is 6; y-intercept is -3

Guided Practice 2: Find Intercepts from a Graph

Find the x- and y-intercepts of the graph.

F x-intercept is 0; y-intercept is 150 G x-intercept is 150; y-intercept is 0

H x-intercept is 150; no y-intercept J No x-intercept; y-intercept is 150

Real-World Example 3: Find Intercepts from a Table

It takes 4 tickets to ride a ride at the zoo. The table shows the function relating the number of tickets that Gavin has andthe number of rides that he has ridden.

O

y.int ( 0,6 )O X - int ( -3,0 )

-

Y - int 101150

-

X - int Now

-

a. Find the x- and y-intercepts of the graph of the function.

b. Describe what the intercepts mean in terms of this situation.

Guided Practice 3: Find Intercepts from a TableThe table shows the function relating the distance to an amusement park in miles and the time in hours the Torres familyhas driven. Find the x- and y-intercepts. Describe what the intercepts mean in this situation.

Example 4: Graph by Using Intercepts and Table of Values

Graph 5x – y = 8 by using the x- and y-intercepts and by using a table of values.

- •

•

•

•

•

-•

( 5,0 )↳

( 0,203

Gavin went on 5 rides before he ran out Of

ticketsGavin had 20 tickets to start .

if - Mt :L 0,248 )

- They started out 248 miles

away S

X - int :#-- It took 4 hrs to get there .

⇒t

1.6 051 D- y=8

5×-0=85zx=8g x=

't 1 -3 5 - y=8y - int 0 -8

- 5 -5

510 ) - y =8y=

-8 +y=-3• t .

- \

- y=8 a

Guided Practice 4: Graph by Using Intercepts and Table of Values

Graph each equation by using the x- and y-intercepts and by using a table of values.

a. –x + 2y = 3 b.

Example 5: Graph by Using Table of Values

a. y=-2 b. x=3

Lesson 3-2 Solving Linear Equations by Graphing

Objectives: I can solve linear equations by graphing.

I can estimate solutions to a linear equation by graphing.

•

•

¥9 x.int- 5

0 -50=4+5

+5

5 :#a -1

ht

¥5t- ( 07+29=3

(-5107

-3 0 2y=3' '

t z I ytnt•

•y= 1,5

•

y=- 0-5

× - int.

4=-5•

- × # =3

-

×=3×= -3TT

•

•

•

• • • •

•

•

CCSS: A.REI.10, F.IF.7a, MP.4

Example 1: Solve an Equation with One Root

Solve by graphing. Verify your solution algebraically.

Guided Practice 1: Solve an Equation with One RootSolve the equation by graphing. Verify your solution using algebra.

Example 2: Solve an Equation with No Solution

a. Solve by graphing and algebraically: 2x + 5 = 2x + 3

b. Solve graphically: (matches video if more practice needed)

.

- Algebra

X - 2=-6 X -256•

the+6 +2+2

••

×+y=o ×= -4EM

. .

: --

¥074 -- 4 ( 014 )1+4=5 L 1/5 )3 | 3+4=7 ( 3) 7)

ii.. yAlgebgq••0*51016=6

0=25×+6

(-15,5 siois ...

5 ¥15 ) -16=8-

6.gs' 6

/ 10 ¥uo)+6=|o -6=23 'xE15

|2z(is )+6 - 12 -30

y Yz =×

-, -15=1/-0

; ÷ ¥3,j## Notion- -

/ NosowiiunSol

EHIME.tt?stEI%aE#tEEs*HIY*nomm

Real-World Example 3: Estimate by GraphingA caterer paid $240 for food for a party. She is charging $15 per person. The function P = 15n – 240 represents thecaterer’s profit P if n people attend the party. Find the zero and explain what it means in the context of this situation.

Guided Practice 3: Estimate by GraphingAntoine’s class is selling candy to raise money for a class trip. They paid $45 for the candy, and they are selling eachcandy bar for $1.50. The function y = 1.50x – 45 represents their profit y when they sell x candy bars. Find the zero anddescribe what it means in the context of this situation.

-

.

o

0 = 15h - 240

¥¥IkI:÷;÷÷÷¥÷"*ii*

n=l6 → WYTILL a

profitof

mum

Lesson 3-3 Rate of Change and Slope

Objectives: I can use rate of change to solve problems.

I can find the slope of a line.

CCSS: F.IF.6, F.LE.1a, MP.2

Real-World Example 1: Find Rate of Change

Use the table to find the rate of change. Then explain its meaning.

Guided Practice 1: Find Rate of ChangeThe table shows how the tiled surface area changes with the number of floor tiles. Find the rate of change. Explain themeaning of the rate of change.

Real World Example 2: Compare Rates of ChangeThe graph below shows the density of population for the state of Idaho in various years.

rise

Fur

22.50

-3=187.5032> 22.50 dof > 22.50 price

of a

ticket

48 →the

area3L> 48 -3=16 otatiles

ZL > 48 helsinz

a. Find the rates of change for 1930-1960 and 1990-2000.

b. Explain the meaning of the rate of change in each case.

c. How are the different rates of change shown on the graph?

Guided Practice 2: Compare Rates of ChangeRefer to the graph below. Without calculating, find the 2-year period that has the least rate of change. Then calculate toverify your answer.

Example 3: Constant Rates of ChangeDetermine whether each function is linear. Explain.

a.

30499368/58.4>2.7 3o±= -09 a Ltd!f11,251>3.4 3¥ ."

gained . oq

peopleeach Year between 1930 -1960

, they Wmiles

each year between 1990 - 2000, They gained -34 peoppfem,↳<°a" War

on averagea-

the steeper the line the bigger therate of change .

zooz-2004

÷± .

gzooo -2°02

f- = 3.5

zoo4w°"

1-02=5

Yes,

it's linear bkthe rates of change are

I < > 5 Constant

it> s 5T = I = ,I=}

, < > 5

i< > 5

b. Guided Practice 3: Constant Rates of ChangeDetermine whether each function is linear. Explain.

a.

b.

Not kmarbk

rates of change are

not constant.

y< > 2 } I I4< > 3

4< > 4

4L > 5

i<> 4 k¥ :# if

12> 4

z<> 4 NO linear

|<> 4

Yes it is-3L> 5 linear

-3 < > 5the rate or

-

3<> 5 change

-3C> 5

µg±gstanHy

Example 4: Positive, Negative and Zero Slope

Find the slope of a line that passes through each pair of points.

a. (-1, 1) and (2, 2)

b. (1, 3) and (4, 1)

c. (-3, 6) and (4, 6)

Guided Practice 4: Positive, Negative andZero SlopeFind the slope of a line that passes through eachpair of points.

a. (-4, 2) and (-2, 10)

b. (6, 7) and (-2, 7)

c. (-2, 2) and (-6, 4)

Example 5: Undefined SlopeFind the slope of a line that passes through (-3, 5.5) and (-3, 2.5)

• . • :" " " "

"x¥x,

= 5¥ :$→•±;→- 2/3

X , Y , Kz Yz

6+4=0-7--0

;fog. . . •

F=4- y?=0?÷z . ÷=÷

-

55=23.52 = T undefined !

Guided Practice 5: Undefined SlopeFind the slope of a line that passes through (6, 3)and (6, 7).

Example 6: Find Coordinates Given the SlopeFind the value of r so that the line through (-6, -3) and (-1, r) has a slope of 2.

Guided Practice 6: Find Coordinates Given the SlopeFind the value of r so that the line that passes through each pair of points has the given slope.

a. (-2, 6), (r, -4); m = -5

b. (r, -6), (5, -8); m = -8

×, Y , xz yz Th

Yz - Yim= - 2=Yt¥ }¥¥Yield :3 '

Xz - X, -3 3

7€X

, Y , Xzyz on

.

.

→ =I¥¥¥'¥

.= -54+4=-1047 ¥r=¥-

5r - 10 = - 10 @X, y , Xz 52 m

+10 +10

-8

the-8 = F- z=

- 8C5)+tDtr)

-0- 2=-40 -1 8r÷=Xs÷+3¥.gl?04@