Unit 4 Day 1 - Slope Rate of...

Unit 4 – Line Equations

Day 1 - Slope Rate of Change

Objectives: SWBAT find the slope of a line or two points.

SWBAT find the rate of change.

Slope of a line - rate of change of a line

Words: rise up or down

mrun right

Formula: 2 1

2 1

y ym

x x

a. Find the slope of line a. b. Find the slope of line b.

From point to point line a goes from point to point line a goes

1 1

2 2

upm

right

2 2

1 1

downm

right

Types of Slope

Positive Slope (goes up) Negative Slope (Goes

down) Zero Slope (flat) Undefined (falling)

a

b

Find the slope of the line connecting each pair of points.

c. A(3, 2) and B(0, -4)

Graph the line drawn by connecting each pair of points, then find the slope using the graph and formula.

d. P(-5, 1) and Q(1, -3) e. C(-1, 3) and D(4, 3) f. P(1, -5) and Q(1, -3)

Athletic Supporters: In 2001, a survey of Reno High students showed that 1020 students attended at least one

sporting event during the year. In 2011, a similar survey showed that 1400 of students had attended at least one

sporting event. At what rate is the support for athletics increasing?

1 1,x y 2 2,x y

2 1

2 1

4 2 6 2 2

0 3 3 1

y ym or

x x

2 1

2 1

3 1 4 2

1 5 6 3

y ym

x x

2 1

2 1

3 3 0

4 1 5

y ym Zero

x x

2 1

2 1

3 5 2

1 1 0

y ym Undefined

x x

1400 1020 380 38 38

2011 2001 10 1m rate of change or students per year

Unit 4 – Day 2 – Graph Using Slope-Intercept Form

Objectives: SWBAT graph a line using slope intercept form

SWBAT find the slope and y – intercept

Slope-intercept form - y mx b m = slope (usually written as a fraction)

b = y intercept (crosses the y axis)

Standard Form – Ax By C (A and B are constants and C cannot equal Zero)

Write an equation in slope-intercept form using the following information.

a. a slope of 2 and a y-intercept of 6 b. a slope of 1 and a y-intercept of 3

2

int 6

2 6

2 6

slope m

y ercept b

y mx b

y x

y x

1

int 3

1 3

3

slope m

y ercept b

y mx b

y x

y x

c. d.

1 11

1 1

int 0

1 0

rise downslope m

run right

y ercept crosses the y axis b

y mx b

y x

y x

3 33

1 1

int 1

3 1

3 1

rise upslope m

run right


y mx b

y x

y x

e. 4x – 2 = 2y – 2 f. 3x – 5 = 4

Solve for y Since there is no y, solve for the x

i. a slope of 2 and passes through (4, 3) j. passes through (6, 5) and has a slope of 2

3

Since we are missing the y intercept, plug in

The point into the slope intercept formula, an

Solve for b, then re-write the formula

k. passes through (-9, 5) and has a slope of 1 l. passes through (8, 3) and (0, 7)

In this example, we are missing the slope and the

y intercept. Use the slope formula to find the slope

then, use the slope intercept formula again to find

the y intercept, and lastly re-write the formula.

3 5 4

5 5

3 9

3 9

3 3

3

x

x

x

x

4 2 2 2

2 2

4 2

4 2

2 2

2

x y

x y

x y

y x

,x y

2

3 2 4

3 8

8 8

5

slope m

y mx b

b

b

b

2

5

2 5

2 5

m

b

y mx b

y x

y x

2

3

25 6

3

5 4

4 4

1

slope m

y mx b

b

b

b

2

3

1

21

3

21

3

m

b

y mx b

y x

y x

1

5 1 9

5 9

9 9

14

slope m

y mx b

b

b

b

1

14

1 14

14

m

b

y mx b

y x

y x

2 1

2 1

7 3 4 1

0 8 8 2

y ym

x x

,x y

,x y 1 1,x y 2 2,x y

1

2

13 8

2

3 4

4 4

7

slope m

y mx b

b

b

b

1

2

7

17

2

17

2

m

b

y mx b

y x

y x

Day 3 – Write Linear Equations in Point-Slope Form

Objectives: SWBAT find the equation of a line that passes through one or two points.

Point-slope form of a linear equation – h and k Point Slope Version –

1 1

1 1, int

y y m x x

m slope

x y a po

, int

y m x h k

m slope

h k a po

You can use either formula, decided which one you like best, and always use that one

Write an equation in point-slope form using the following information. Then change the equation to slope-

intercept form.

a. passes through (3, 7) and has a slope of 4 b. has a slope of 1

2 and passes through (0, 6)

c. has a slope of 5 and passes through (2, 0) d. passes through (4, 1) and (4, 2)

1 1,x y

1 1,x y

,h k

1 1

1 1

4

, int 3,7

7 4 3

7 4 3

7 4 12

7 7

4 19

y y m x x

m slope

x y po

y x

y x

y x

y x

1 1

1 1

5

, int 2,0

0 5 2

0 5 2

5 10

y y m x x

m slope

x y po

y x

y x

y x

1

2

, int 0,6

10 6

2

16

2

16

2

y m x h k

m slope

h k po

y x

y x

y x

2 1

2 1

2 1 3

4 4 8

y ym

x x

3

8

, int 4,2

34 2

8

3 36

8 2

3 9

8 2

y m x h k

m slope

h k po

y x

y x

y x

e. f. Graph: y – 4 = 3(x – 1)

Point two points on the graph then find the

slope, and then the equation.

(−𝟏, −𝟐) & (𝟐, −𝟏)

1 1

1 1

1

4

, int 1, 2

12 1

4

12 1

4

1 7

4 4

y y m x x

m slope

x y po

y x

y x

y x

2 1

2 1

1 2 1

2 2 4

y ym

x x

4 3 1

4 3 3

4 4

3 1

y x

y x

y x

3 33

1 1

int 1

rise upslope m

run right


Day 4 – Write Equations of Parallel

Objective: SWBAT write the equation of lines parallel to other lines.

Slopes of Parallel Lines – Lines that don’t intersect because they have the same slope

Slopes of Perpendicular Lines – Lines that intersect at a 90 degree angle because their slopes are opposite reciprocals.

Determine if each pair of lines are parallel, perpendicular, the same line, or intersecting.

Find the slope of each line,

if the slopes are the same parallel

if the slopes are the same and the y intercept are the same same line

if the slopes are opposite reciprocals (opposite signs, and flipped fraction) perpendicular

anything else the lines are just intersecting.

a. 1

3y x and 53y x b. 2 3y x and

23

1y x

Perpendicular Intersecting

c. 3y x and 1y x d. 4

33

y x and 4

35y x

Parallel Perpendicular

Writing equations of PARALLEL lines.

i. Write the linear equation parallel to y = 4x + 1 j. Write the equation of the line parallel to

and passes through (2, 4). 4x – 2y = 10 that passes through (-6, 1).

1 1

1 1

4

, int 2,4

4 4 2

4 4 2

4 4 8

4 4

4 4

y y m x x

m slope

x y po

y x

y x

y x

y x

4 1

4

parallel lines same slope

y x

m

4 2 10

4 4

2 4 10

2 4 10

2 2

2 5

2 5

2

x y

x x

y x

y x

y x


y x

m

2

, int 6,1

2 6 1

2 6 1

2 12 1

2 13

y m x h k

m slope

h k po

y x

y x

y x

y x

g. Which equation of the line passes through (8, 10) and is parallel to the graph of

the line 𝑦 = 8

3𝑥 + 7 .

h. Write the equation of a line that passes through the origin and is parallel to the line 2𝑦 − 6𝑥 = 4.

1 1

1 1

8

3

, int 8,10

810 8

3

810 8

3

8 6410

3 3

10 10

8 64 30

3 3 3

8 34

3 3

y y m x x

m slope

x y po

y x

y x

y x

y x

y x

87

3

8

3


y x

m

2 6 4

6 6

2 6 4

2 6 4

2 2

3 2

3 2

3

y x

x x

y x

y x

y x


y x

m

3

, int 0

3 0 0

3

3

y m x h k

m slope

h k po

y x

y x

y x

Day 5 – Write Equations of Perpendicular Lines

Objective: SWBAT write the equation of lines perpendicular to other lines.

Slopes of Parallel Lines – Lines that don’t intersect because they have the same slope

Slopes of Perpendicular Lines – Lines that intersect at a 90 degree angle because their slopes are opposite reciprocals.

Determine if each pair of lines are parallel, perpendicular, the same line, or intersecting.

Find the slope of each line,

if the slopes are the same parallel

if the slopes are the same and the y intercept are the same same line

if the slopes are opposite reciprocals (opposite signs, and flipped fraction) perpendicular

anything else the lines are just intersecting.

a. 3 5y x and 5 3y x b. 3y x and 3y x

Same Line Perpendicular

c. 4 3x y and 5 6x y x d. 2 6 2y x and 3 0x y

4 3

3 3

4 3

x y

y x

5 6

5

4

5

6

x y x

x x

y x

2 6 2

3

2 2

1

y x

y x

3 0

3

3

3

x y

y y

y x

Intersecting Parallel

e. Which statement is true for the given lines?

Line a: 3𝑥 − 4𝑦 = −4 Line b: 3𝑥 − 4𝑦 = 8 Line c: 3𝑥 + 4𝑦 = 8

A. Lines a and c are parallel B. Lines a and b are parallel

C. Lines b and c are parallel D. Lines a and c are perpendicular

E. Lines a and b are perpendicular

Writing equations of PERPENDICULAR lines.

f. Write the equation of the line perpendicular to g. Write the equation of the line that passes through

y = 2x – 5 that passes through (6, 7) (–7, –3) and is perpendicular to x – 3y = 5.

3 4 4

31

4

x y

y x

3 4 4

32

4

x y

y x

3 4 4

32

4

x y

y x

1 1

1 1

1

2

, int 6,7

17 6

2

17 6

2

17 3

2

7 7

110

2

y y m x x

m slope

x y po

y x

y x

y x

y x

2 5

1

2

perpendicular lines oppsoite recipricol

y x

m

3 5

3 5

3 5

3 3

1 5

3 3

1 5

3 3

3

x y

x x

y x

y x

y x


y x

m

3

, int 7, 3

3 7 3

3 7 3

3 21 3

3 24

y m x h k

m slope

h k po

y x

y x

y x

y x

h. Write the equation of a line that passes through the origin and is perpendicular to the line 𝑦 − 6 = 0.5𝑥

i. Which equation of the line passes through (4,7) and is perpendicular to the graph of the

line that passes through the points (1,3) and (−2,9) ?

6 0.5

6 6

0.5 6

0.5 6

16

2

2

y x

y x


y x

y x

m

2

, int 0,0

2 0

2

2

y m x h k

m slope

h k po

y x

y x

y x

2 1

2 1

9 3 62

2 1 3

y ym

x x

2

, int 1,3

2 1 3

2 2 3

2 5

y m x h k

m slope

h k po

y x

y x

y x

1 1

1 1

1

2

, int 4,7

17 4

2

17 4

2

17 2

2

7 7

15

2

y y m x x

m slope

x y po

y x

y x

y x

y x

2 5

1

2


y x

m

Distance between Points and Lines – Day 1

Objectives: SWBAT find the distance between parallel lines

Find the distance between the point and the line.

Count the points horizontally for vertically

a. b. c.

3 units 9 units 2 units

d. 𝑥 = 2 and (4,5) e. 𝑦 = 3 and the origin

When finding the distance between the point and vertical/horizontal line, you counted the distance or length of the ___Perpendicular__________ line.

Distance between a Point and a line.

Find the perpendicular slope to your line

Write the equation of a line is perpendicular to the line and goes through the

indicated point

Set two lines equal to each other

Find the solution (x and y)

Find the distance between your x and y and the given point.

f. Find the distance between the line 𝑦 = −𝑥 − 2 and the point (2, 4).

Find ⊥ slope Find ⊥ line equation through

the point Set two equations equal

Solve for x and y Distance Formula

g. Find the distance between the line 𝑦 = 2𝑥 + 5 and the point (7,-1).




11

1

1

1

m

m

1 Point 2, 4

1 2 4

2 4

2

y m x h k

m

y x

y x

y x

2

2

2 2

y x

y x

x x

2 2

2 2 2

2 2

2 4

2

x x

x x

x

x

x

2

2

2 2

0

y x

x

y

y

2 2

2 1 2 1

1 1 2 2

2 2

2 2

, 2, 4 , 2,0

2 2 0 4

4 4

16 16

32

4 2

d x x y y

x y x y

d

d

d

d

d units

22

1

1

2

m

m

1 Point 7, 1

2

17 1

2

1 71

2 2

1 5

2 2

y m x h k

m

y x

y x

y x

2 5

1 5

2 2

1 52 5

2 2

y x

y x

x x

1 52 5

2 2

1 1

2 2

5 55

2 2

5 5

5 5

2 2

1

x x

x x

x

x

x

2 5

1

2 1 5

3

y x

x

y

y

2 2

2 1 2 1

1 1 2 2

22

2 2

, 7, 1 , 1,3

1 7 3 1

8 4

64 16

80

4 5

d x x y y

x y x y

d

d

d

d

d units

h. Find the distance between the line 𝑦 =1

2𝑥 + 1 and the point (3,10).




1

2

2

m

m

2 Point 3,10

2 3 10

2 6 10

2 16

y m x h k

m

y x

y x

y x

11

2

2 16

11 2 16

2

y x

y x

x x

11 2 16

2

2 2

51 16

2

1 1

515

2

6

x x

x x

x

x

x

11

2

6

16 1

2

4

y x

x

y

y

2 2

2 1 2 1

1 1 2 2

2 2

2 2

, 3,10 , 6, 4

6 3 4 10

3 6

9 36

45

3 5

d x x y y

x y x y

d

d

d

d

d units

Unit 4 Day 1 - Slope Rate of...

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