Chapter 4

Analyzing Linear Equations

4.1 Rate of Change & Slope

• Rate of Change- a ratio that describes, on average, how much one quantity changes compared to another

Rate of change=

Change in y

Change in x

Ex:

Time

Walking(sec)

Distance

Walked(ft)

1 4

2 8

3 12

4 16

x y

+1

+1

+1

+4

+4

+4

Rate of change =

4

1

= 4 ft/sec

Negative slope Positive slope

Zero slope Undefined slope

• Slope- the ratio of the change in y-coordinates over the change in x-coordinates for a line

5

4

5

4

41

62

m

Slope=rise

run

Slope:

m =y2 – y1

x2 – x1

Ex: Ex:Find the slope (4, 6) (-1, 2)x1 y1 x2 y2

11

10

65

55

m

Ex: Find the value of r so that the line through (10, r) and (3, 4) has a slope of -2/7 x1 y1 x2 y2

103

4

7

2

r

)4(7)7(2 rr72814

-28 -28-14= -7r/-7 /-7

r = 2Plug in the m and the coordinates for the slope

Simplify and solve for r

4.2 Slope and Direct Variation

• Direct Variation- a proportional relationshipy = kx

*this represents a constant rate of change and k is the constant of variation

Ex:a. Name the constant of variation

b. Find the slope

c. Compare slope and the constant of variation. What do you notice?

k = -1/2

They are the same thing

2

1

4

2

04

02

m

• Graph a direct Variation:

1. Write the slope as a ratio

2. Start at the origin (0, 0)

3. Move up and across according to the slope

4. Draw a line to connect the ordered pairs

1

4m

x

y

a. y = 4x b. y = -1/3x

Up 4 and right 13

1m

Down 1 and right 3

4.3 Slope-Intercept Form

y = mx + b

slope y-intercept

Ex: Write an equation in slope-intercept form given slope and the y-intercept

Slope = 3 y-intercept = -5

y = 3x - 5

a. b. y-intercept= (0, 6) m = 2/5

y = 2/5x + 6

Ex:

Ex:

Write an equation in slope-intercept form given a graph.

Graph an equation given in slope-intercept form

Graph from an equation:

1. write the slope as a ratio and y-intercept as an ordered pair

2. Graph the y-intercept

3. Use slope to find other points and connect with a line3

5

30

32

m

y-intercept= (0, 2)

So b=2

y = 5/3x + 2

x

y

a. y = -2/3x + 1

b. 5x – 3y = 6

m = -2/3 y-int= (0, 1)

5x – 3y = 6

-5x -5x

-3y = -5x + 6

/-3 /-3

y = 5/3x - 2 m = 5/3 y-int= (0, -2)

6.7 Graphing Inequalities with Two Variables

• The equation makes the line to define the boundary• The shaded region is the half-plane

1. Get the equation into slope-intercept form2. List the intercept as an ordered-pair and the slope as a

ratio3. Graph the intercept and use the slope to find at least 2

more points4. Draw the line (dotted or solid)5. Test an ordered-pair not on the line

1. If it is true shade that side of the line2. If it is false shade the other side of the line

< or > or

Dotted Line Solid Line

Ex1: y 2x - 3

m =

b = -3 = (0, -3)1

2

Use a solid line because it is

Test: (0, 0)

0 2(0) – 3

0 0 – 3

0 -3 false (shade other side)

• Ex2: y – 2x < 4

y – 2x < 4

+ 2x +2x

y < 2x + 4

m =

b = 4 = (0, 4)1

2

Test: (0, 0)

0 < 2(0) + 4

0 < 0 + 4

0 < 4 true (shade this side)

Use a dotted line because it is <

• Ex3: 3y - 2 > -x + 7

3

1

3y – 2 > -x + 7

+2 +2

3y > -x + 9

/3 /3 /3

y > - x + 3

m = -

b = 3 = (0, 3)3

1

Test: (0, 0)

0 > - (0) + 3

0 > 0 + 3

0 > 3 false (shade other side)

3

1

Use a dotted line because it is >

4.4 Writing Equations in Slope-Intercept Form

y = mx + b

slope y-intercept

Ex. Write an equation given slope and one point

Write the equation of the line that passes through (1, 5) with slope 2

y = mx + b

y = 2x + b

5 = 2(1) + b

5 = 2 + b

-2 -2 3 = b

y = 2x + 3

Replace m with the slope

Replace the x and y with the ordered pair coordinates

Solve for b (the y-intercept)

Replace the numbers for slope and the y-intercept

Ex. Write an equation given two points

Write an equation for the line that passes through (-3, -1) and (6, -4)

y = mx + b

y = -1/3x + b

3

1

9

3

36

14

m

-1 = -1/3(-3) + b

-1 = 1 + b-1 -1

-2 = b

y = -1/3x - 2

Find slope

Replace m with the slope

Replace the x and y with one of the ordered pair coordinates

Solve for b (the y-intercept)

Replace the numbers for slope and the y-intercept

4.5 Writing Equations in Point-Slope Form

Point-Slope Form: y – y1 = m(x – x1)

Ex:

Ex:

Write an equation given slope and one point

y – y1 = m(x – x1)Write the equation of a line that passes through (6, -2) with slope 5

y – -2= 5(x – 6)y + 2 = 5(x – 6)

Replace m with the slope and y1 and x1 with the ordered pair

Simplify

Write an equation of a horizontal line

y – y1 = m(x – x1)y – 2 = 0(x – 3)y – 2 =0

Replace m with the slope and y1 and x1 with the ordered pair

Simplify

y = 2

Write the equation for the line that passes through (3, 2) and is horizontal

Ex: Ex:

Ex:

Write an equation in Standard Form

Write an equation in Slope-Intercept Form

Write an equation in Point-Slope Form given two points

y + 5= -5/4(x -2)

4[y + 5= -5/4(x -2)]4y + 20= -5(x -2)

4y + 20= -5x + 10+5x +5x

5x + 4y + 20= 10

- 20 -205x + 4y = -10

Multiply by 4Distribute

Add 5x(move it to the other side)

Subtract 20(move it to the other side)

Simplify

y -2= ½(x+ 5)2[y -2= ½(x+ 5)]2y -4= x+ 5 +4= +42y = x+ 9/2 /2y = 1/2x+ 9/2

Multiply by 2

Add 4(move it to the other side)

Divide by 2

Simplify

Write the equation for the line that passes through (2, 1) (6, 4)

4

3

26

14

m

y - 1= 3/4(x -2)

y - 4= 3/4(x -6)or

Find slope

Replace m with slope and y1 and x1 with one of the ordered pairs

4.6 Statistics: Scatter Plots and Lines of Fit

• Scatter Plot- a graph in which two sets of data are plotted as ordered pairs in a coordinate plane – Used to investigate a relationship between

two quantities

Positive Correlation

Negative Correlation

No Correlation

• If the data points do not lie in a line, but are close to making a line you can draw a Line of Fit– This line describes the trend of the data (once you have this line

you can use ordered pairs from it to write an equation)

Ex:

years 1 3 5 8 12 12 13 15

Vertical Drop

151 156 225 230 306 300 255 400

a. Make a scatter plot b. Draw a line of fit. What correlation do you find?

c. Write an equation in slope-intercept form for the line

Positive correlation

1 3 5 7 9 11 13 15

100

200

300

400

500

(8, 230) (12, 306) Find slope

m= 19

Plug in an ordered pair and slope to find the y-intercept

y = 19x +78

4.7 Geometry : Parallel and Perpendicular Lines

• Parallel lines- do not intersect and have the same slope

• Perpendicular lines- make right angles and have opposite slopes

a

b Line a has slope 6/5

Line b has slope 6/5

c

d

Line c has slope 3/2

Line d has slope -2/3