NOTES: Graphing Linear Equations

Label the following:

a. x-axis

b. y-axis

c. origin

d. 4 quadrants

e. K(4,2)

r L (0,-3)

h. P(-1,5)

Plot the following points:

/

: i

!

i

List all the words that stand for the following variables:

X

Y

Graph the equation:

1) Solve for y if needed.

2) Set your own domain and plug in the input to solve for the

output in the process column.

3) GrEo_pÿh the solution set. (the set of ordered pairs)

1. y=2x+2

input y=2x+2 output (X, y)

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f

2. 2x- 2y = -6

domain y= ÿffo% range (x , y)

. Farmer Cheng has a pocket full of quarters and dimes worth $2.50. If he has no dimes, how manyquarters does he have? If he has no quarters, how many dimes does he have? If he has 10 dimes,

how many quarters does he have? ®ÿ1ÿ%\ÿ (ÿw ÿ0ÿ(ÿ

I quartersdimes

\0 I o to0I /ÿ 10

Graph your findings. Don't forget to label your axis!

_ÿ,, o

\¢)6o¢d. .

_ ° ÿ \ \A°,_

.¢ÿ\.% .Zg" ÿ

8. it, take.s oÿe mhuÿe to fill a fonr-1ÿaHon e, ont-ah:ÿer

aI: the I!:-xÿ,,(er spring, iiow |ong- does it tÿq.:e to fiH a 20

1.5

] (}

"r,

m---

2

,i,,,'iii jd LJ-L..

x : : :

: : : : : :

When the coefficient of x is a fractionÿ choose x-values

that are multiples of the denominator.

3. 2x+3y=12

independent dependent (x, y)

!I!

,r. I

i ....

4. y = -2

!

I

domain y = ÿ ÿ range (x , y)

5. X=3

input x = output (x, y)I

Slope:

9. Each step of the stairs leading from room 9 to room 107 in a house has a vertical rise of 7 inches and a horizontal run

of 12 inches. Each step of the marble staircase leading to the hall has a vertical rise of 5.5 inches and a horizontal run of

13 inches.

a) Which flight of stairs do you think is steeper? Why

b) Calculate the ratio rise/run for each flight of stairs, and verify that the greater ratio belongs to the flight you thought Ato be steeper. ÿ<oÿ ÿ-ÿ,ÿ ÿ,,ÿ(ÿ ÿ.ÿ ) ÿJ¢ ÿrÿ,ÿf. ÿ ÿ Coÿÿÿ"

/

10. The slope of a line is a measure of how steep the line is. It is calculated by dividing the change in the

y-coordinates by the corresponding change in the x-coordinates between two points on the line:

slope- changeChange inxin y Q)f ÿ ÿ I 5 ÿ(ÿ(0 "C-ÿ("ÿ(ÿÿ e.ÿ

Calculate the slope of the line that goes through the two points (1, 3) and (7, 6).I •

b) Calculate the slope of the line that goes through the two points (0, O) and (9, 6).

UsinÿJÿpe = change iny find the slopes of the lines represented by the tables below.change in x p

11. The table shows the cost of using a computer at an Internet cafÿ for a given amount of time. Find the rate of

change in cost with respect to time. ÿ

j (.O:it {1011ii'I}

2 4 , 6 1: ......... ÿ::

7 14 2"1

¢ 1.---. o( 5£

12. The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time.

, ! ( i

oÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿoooÿÿÿÿÿeeÿÿÿÿÿÿÿÿÿÿeÿÿÿÿÿoÿÿÿÿÿÿÿÿÿÿeÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿe

a

To find the slope given two points.oaa

oo

oooa

aa

a

1. Plug into the formula. Y2- Yl = change in ym=x2-x1 change inx

2. Reduce if needed.

o.ÿoÿoÿeÿeoÿeÿeÿoÿ...ÿ.ÿo...eÿo.eÿ.ÿ.ÿe..ÿ$ÿ.ÿeo.oÿeÿ.ÿ.ÿoÿ.

Find the slope of a line given two points.

13. (9,2) and (6,5) 14. (4,-5) and (-2,-7) 15. (-2,3) and (-4,-3)

fGiven the slope of a line+ find the missing coordinate.

(2, 3) and (x, 9) has a slope of a2

17. Find the value of y so that the line that passes through the point

(0, -3) and (-9, y) has a slope of--ÿ.

.o, ma\ V\vt

16. Find the value of x so that the line that passes through the point

J

FOUR TYPES OF SLOPE

SLOPE (m): the steepness of a line; the constant rate of change

change in y rise

change in x run

Jnegative Zero positive

slope slope slope

undefined

slope

Plot the points, draw the line, identify the type of slope and calculate the

actual slopÿ.

18. (-2,3)and (4,-5) 19. (4,-1)and (2,-1) 20. (-2,0) and (0,5)

\

mÿmFmEÿ m\

J

f

/

7Aÿo %ÿ9e

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

1. Pick 2 points

2. COUnt riserun

21. 22. 23.

j!

i \.,

f

r p, "ÿ

m m m

24. 25.

ffm ÿjr

Lÿ

I77,---- 117, -- --

26. Explain why the descriptions "right 5 up 2", "right 10 up 4", "left 5 down 2", "right 5/2 up 1", and "left

1 down 2/5" all describe the same inclination for a straight line.

1

<- v

9_ 9_.

27.

What do you notice?Draw the segment from (3, 1) to (5, 6), and the segment from (0"5) to (2, 0). Calculate their slopes.

Which line is steeper?

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f

Parent Function:

Graph the following function:

y=X

ff,

i(/,

/

a. What is the slope of this line?

1 3y=2x y =-3x y= ÿx y= -ÿx

I I\

! \ i \.

I \ ÿ "ÿ; \

Graph the following functions.

b. Where does the line cross the y-axis? 0ÿ¥ (ÿ

c. Define y-intercept.

d. This type of line is called a PARENT FUNCTION. Describe two

characteristics of a parent function. (HINT: slope? y-intercept?)

a. What do you notice when you change the coefficient of x?

Cÿ "ca" ÿcÿcÿcÿ oÿÿ ÿ ÿ,ÿ ÿ\ÿ9ÿ c5ÿ ÿ

b. What does changing the coefficient to a whole number do?

c. What does changing the coefficient to a fraction do?

d. What is the difference between a positive and negative coefficient?

e. What does the coefficienÿ represent.

Graph the following functions.

y=x+2

! //

/

/

y= X--3

J,/

,I

I

//

==i/o

a° What do you notice when you change the constant?

b. What does changing the constant to a positive number do?

c. What does changing the constant to a negative number do?

d. What does the constant represent?

identify the slope and y-intercept of the following equations:

1. y=2x+3

Slope ...... y- intercept = $

-42. y=--x-6

7

Slope = ÿ"qt°7 y-intercept : ÿ ÿD

3. y = 120x + 300

Slope = ÿ ÿ)1

y- intercept =

2 34. y'-x--

3 4

Slope = y- intercept =

If the slope-intercept form of a linear equation is y :mx "Jr" b then the

b ,_.

Linear parent function: f (x) = x.

® When you change the slope (m) of a linear equation, it will affect the slopeof the line.

® The slope of the linear parent function is : m = 1.® If the slope of a line is greater than 1, the line will be steeper.® If the slope of a line is between 0 and 1, the line will be flatter.® If the slope is negative, the direction of the line changes.® When you change the y-intercept (b) of a linear equation, the line will be

translated (shifted up or down the y-axis) from the origin.® The y-intercept of the linear parent function is: b = O.® If the y-intercept of a line is greater than O, the line will be translated up.® If the y-intercept of a line is less than O, the line will be translated down.

.aph the function. Compare the graph with the graph of/(x) = x.

15. f(x) = 3x + s 6. f(x) = - x - 3

27. f(x) =-2x + 3

8. y=4x-2

X y = 4X-2 Y (x, y)[

,i°

F

[

From the graph, identify the x-intercept, y-intercept, and slope.

Slope = ÿ

x-intercept = ÿ/ÿ

y-intercept = ÿgÿ

Linear equations that look like y = mx + b are said to be in slope-intercept form. Explain.

The terminology refers to which of the two intercepts, x or y?

19. y=2x-4 10. Y=2X+3

Slope: ÿ Slope: ÿ]ÿ

y-intercept: ÿ ÿ y-intercept:

/o

fJJA

J!

/f

11. 3x+ 6y = 12

Slope:

y-intercept: ÿ

A

r

12. Chris does a lot of babysitting. When parents drop off their children and Chris can supervise athome, the hourly rate is $3. If Chris has to travel to the child's home, there is a fixed charge of $5 for

transportation in addition to the $3 hourly rate.

a) Graph y = 3x and y = 3x + 5. What do these lines have to do with the babysitting context? What

feature do they have in common? How do they differ?

F

/°

b)

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Predict what the graph ofy = 3x -I- 6 will look like. What change in the babysitting context does this line suggest?

u n

: Slope-Intercept Form: y = mx + b OR y = b + mx !g mg Rm R

: Function Notation: f(x) = mx + bn m

Ru mnm m

i The symbol f(x)is just another name for y. We read it as "the im

i function of x"! in o

u u u m u u n m u m m u m n N R n n n B nm i l u l n n n m n n n i u n u m u n u u m u u m u u m inn n n i n muB mll B m R m n n m u n i u i n n u m nu B n n u u u i un m u u i H is m n m u mm n m i u u m Bun m u o R in l l n mln n nn i u n a mÿ ÿ

f (x) : 3x + 4 x f(x)

,

14. Tell whether the graphs of the two equations are parallel lines. Explain your reasoning.

a. Y = 2X + 3; -x + 2y =12 b. 2x + y = 8; Y = -2x + 4 ÿ ÿ°

15. Find the value ofx so that the lines through the given points are parallel.

Line 1: (3, 5) and (0, O)

Line 2: (-2, 6) and (x, 16)

Point-Slope Form:

The equation y-yl=m(x-xl)isinpoint-slopeform.

b) What information did you use to set up this problem?

c) Why do you think it is called point-slope form?

Write an equation of a line in point-slope form given a point and the slope.

1[ \1 ,a, 5, m=-- 2

\ ] 2(-4, 8) m = 6

53. (-10,-1) m=-

6

4. What do the lines Y = 3(x - 1) + 5, y = 2(x - 1) + 5, and

they different?

.ÿ:ÿ(ÿ\) ÿ ÿysLÿ ÿ9

1 -1)+5y =-ÿ(x

I

all have in common. How are

r

.

b

Find an equation in point-slope form for the line containing (-3, 0) and (0, 4). R°o.Z

04% Z

o Write an equation in point-slope form for:

a) the line that goes through (2, 5) and (6, -3).%

b) the line that goes through point (h, k) and that has slope m.

7. The table shows the cost of visiting a working ranch for one day and one night for different numbers of

people.

# of people 4 6 8 10

cost $ iiÿ 250 350 450 550

a) State two ordered pairs from the table.

b) Find the slope/rate of change.(bÿIS6 \66

c) Write an equation in point-slope form.

d) Then convert to slope-intercept form.

Use the equation to find the cost foÿ.

f) How many people could visit for $950?

oso ÿTO ÿ0 50

e)

f

8. Write an equation for the line that contains the points on the table, and make up a word problem for it.

X 0 15 30 45 60

Y 100 160 220 280 340

\ÿ \55 ÿCÿ

14 = ÿ\ÿ

9. Write an equation of a line in slope-intercept form that passes through the

point (8,-3) and has the slope of -5.

Name the slope and one point on the line from each point-slope form equation.

10. y + 5 = -3(x- 9)

12. Answer the questions below to GRAPH the line given in point-slope form.

y+3=l(x+5)-2

a) Name the point.

b) What is the slope?

J1"-

JJ

c) Plot the point on the graph.

d) Then use the slope to determine

two more points on the line.

13. A long distance company charges $0.25 per minute. On Tim's last phone bill he was charged $3.25 (c)

for talking 13 minutes (m). Use the ordered pair (m,c) to write an equation in point-slope form.

® c')

14. If (-3, 5) and (2, 1) are two points on a line, find three other points on the sÿme line.

15. Write the equation of a line that is parallel to the graph of y = 3x - 2 and contains the point (2, -3).

STANDARD FORM: Ax + By = C

X- intercept - where the line crosses the x-axis

(x value when y=0)

Y- intercept - where the line crosses the y-axis

( y value when x=0)

#

J

.J

To find X- and Y- Intercepts

1. To find the y-intercept:2. To find the x-intercept:

plug 0 in for x and solve for y.plug 0 in for y and solve for x.

a

°A

C...

1. 3x-2y = 12 2. y=-3x+3 3. x=2y+4 A -t

%llJll\

ja¥m

\< \

\

L)

¥

a

& ÿvÿ _ I"5

Direct Variation equations can be written in the form y = kx.

The nonzero number k is called the constant of variation, and y is said to vary

directly with x.

These three graphs all represent direct variation. What do they all have in common?

k

\

k

\\

/!/Y

/#

A#!

A

hl

=ÿ

m

Tell whether the equation represents direct variation. If so, identify the constant of variation.

5,-x+y=l 6. 2x+y=O 7. 4x - 5y = 0

g'N'

a !

The data in each table fits a dixect vaiiation. Complete each table, write an equationto model its data, and sketch a graph.

A\(a) :,: '2 4 6 (b) 2 3 5 s

y a 6 cÿ is :ÿ -s -12 -20 °%4

u9% w> oÿ

't\,5

How to write a Direct Variation Equation.

1. Find k: k - -y ÿÿwÿÿ

2. Substitute the value for k into the equation y - kx.

8. Landscapers plan to spread a layer of stone on a path. The number of s bags of stone needed depends on the depth

d (in inches) of the layer. They need 10 bags to spread a layer of stone that is 2 inches deep. Write a direct variation

equation that relates d and s. Then find the number of bags needed to spread a layer that is 3 inches deep.

9. At a recycling center, computers and computer accessories can be recycled for a fee off based on weight w, as shown

in the table.

Weight, w Fee, f(pounds) (dollars)

10 2.50

15 3.75

30 7.50

a. Explain whyfvaries directly with w.

b. Write a direct variation equation that relates w and f. Find the total recycling fee for a computer that weights 18

pounds and a printer that weights 10 pounds.

f

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:!

IZ"!,

i

i

!

,i,

ii!

Solving Equations by Graphing

Graph each line on the same coordinate plane.

1=- _ÿ .... ÿ1. y 2x+5 ÿio

x+y=2 \

-12. y =--x+3

2

. 9-

2x + 4y = 20

3. y=3x-2

3x-y=2 /

What is the solution?

Do they intersect? ÿ ÿ

Where? ÿ\ÿ

Do they intersect? ÿ%ÿ(ÿ

Where?ÿg (ÿ ÿ(ÿ\\ÿ\ \ÿWhat is the solution?1 ÿ(ÿÿ ÿg

Do they intersect? ÿ(ÿ)

What is the solution?

%

To determine which ordered pair is a solution without graphing:

a. Plug the ordered pair (x,y) into the equation.b. If both sides of the equation are equal, then it is a solution

Is a given ordered pair a "solution" to a system of equations??

To be a solution, the ordered pair must work in both equations. Plug in to find out!!!

Show work to prove your answer. State Yes or No.

4) Is (-2, 4) a solution to the system? y = -2x y = x-6

g5) Is (3,-1) a solution to the system? y+2x=5 2 +x=l

Decide which point(s) lie on the graph of the line.

1. x+2y=8 0,5)

tl 5*t4

B. (1,8) (ÿ-2,5)

6)5cÿ ÿsS

The Belton City Parks Department offers a season pass for $45to have access to the city's ÿ.ÿ ÿ ÿ

tennis courts. As a season pass holder, you pay ÿ3 per session for using the city courÿ

Without a season pass, you pay $12 per session for use oYttYe courts.

Writing a System of Equations -which system of equations can be used to find the ÿCÿ

number of sessions of tennis, x, after which the total cost, y, with a season pass, is the same

as the total cost without a season pass?

a) y=3x b) y=3x ÿ)y=45+3x d) y=45+3x

y = 12x y = 45 + 12x y = 12x y = 45 + 12x

Making a Table-Make a table of values

that shows the total cost for a season pass

holder and a non season pass holder after

1, 2, 3, 4, 5, and 6 tennis sessions. After

how many sessions is the cost the same?

# ofSessions

Drawing a Graph - use the table to graph the

system of equations. Under what circumstances

does it make sense to become a season pass

holder? - ÿ \ÿ ÿ\¢ÿ%ÿo5 ¥cÿ M ÿ% k ¥

1

2

3

4

5

6

Cost for Cost forSeason Pass Non- Pass

Holders Holders

15noo

0

\%6

9o /

80 //t

70 /

60 ÿ

5o ÿ/f

/46 "

/30 f

2o ]

lO -4

!

123456789 lO

# of Tennis Sessions