Unit 4 Writing and Graphing Linear Equations

Unit 4

Writing and Graphing

Linear Equations

NAME:______________________ GRADE:_____________________ TEACHER: Ms. Schmidt _

Coordinate Plane

Coordinate Plane

Plot the following points and draw the line they

represent. Write an additional point on the line. 1. (-4,3), (0,2), (4,1) 2. (0,6), (1,4), (6,-6) 3. (0,-2), (2,4), (4,10)

Coordinate Plane

1. Write the coordinate for each of the given points

A F

B G

C H

D

E

Plot the following points and draw the line they

represent. Write an additional coordinate of a point

on the line.

2. (-4,0), (0,3), (4,6), 3. (0,8), (1,6), (2,4), 4.

x y

-3 -5

0 -3

3 -1

5. What is the solution to the equation x + 5 + x + 1 + x = 3(x + 2) + 1

Express each in positive exponential form 7

6. 10

10−7

7. 79

x 75

8. 4-2

x 48

Finding the Slope of a Line Graphically

Vocabulary

Slope: Slope Formula:

Constant of Proportionality

Types of Slopes

1. Positive Slope 2. Negative Slope 3. Zero Slope 4. No Slope/Undefined Slope

Type of Slope= Type of Slope= Type of Slope=

Slope = Slope = Slope =

Type of Slope= Type of Slope= Type of Slope= Slope = Slope = Slope =

6)

Finding the Slope of a Line Graphically Try These:

9)

Slope Slope Slope

Type of Slope Type of Slope Type of Slope

Plot the points and draw a line through the given points. Find the slope of the line.

1. A(-5,4) and B(4,-3) 2. A(4,3) and B(4,-6) 3. A(-3,-2) and B(4,4)

For questions 4-8 determine the type of slope for each of the given lines.

Slope

Slope

Slope

4. 5. 6. 7) 8)

Finding the Slope of a Line Graphically

Plot the points and draw a line through the given points. Find the slope (rate of change) of the line.

1. A(-1,2) and B(-1,-5) 2. A(4,3) and B(-4,-3) 3. A(5,-2) and B(0,4)

4. A(-4,-2) and B(4,-2) 5. A(-5,1) and B( 2,2) 6. A(-3,5) and B(1,1)

7. a) What is the slope of the line shown on the following graph?

b) Explain why you know the slope has that value

8. Solve for x: 4(2x – 6) = 8(x – 3)

Finding the Slope of a Line Graphically

Slope= 𝑹𝒊𝒔𝒆

𝑹𝒖𝒏=

∆𝒙

∆𝒚

Slope is the ratio of the vertical change of the line (difference in y-values) to its horizontal change (difference in x-values). The ratio is a constant rate of change between any two points on the line.

Find the slope (rate of change) of the line containing the following points.

1. (0, 6) and (1, 4) 2. (1, 1) and (2, 4) 3. (3, 5) and (8, 5)

Using the tables below, determine the slope (rate of change) using the slope formula. What is

the domain for each relation? 4. 5.

Find the slope (rate of change) of the line containing the following points algebraically.

6. (-4, 3) and (4, 1) 7. (0, -3) and (5, -1) 8. (-8, 2), (2, -3), and (8, -6)

x y

1 3

2 6

3 9

4 12

x y

8 4

6 8

4 12

2 16

Finding the Slope of a Line Graphically

Using the graphs below, determine the slope. Check by showing it algebraically.

9. 10.

Find the slope of the given table by using the slope formula

11. 12. x y

6 32

12 24

18 16

24 8

x y

2 0

4 3

6 6

8 9

Finding the Slope of a Line Graphically

Given the two points, find the slope of the line algebraically.

1) (2,3) and (5,6) 2.) (-2,-2) and (1,-2) 3) (2,1) and (2,-2)

Using the tables below, determine the slope using the slope formula.

4) 5)

Determine the type of slope of the given

line. 6) 7)

9) Simplify: 3(-4)2 – 12 10) Simplify: 47 ∙ 53 ∙ 42

11) Find the slope of the given line:

8)

x y

3 6

1 8

-1 10

-3 12

x y

0 1

4 3

8 5

12 7

Understanding Slope and Y-Intercept

Where does the line cross the y-axis?

A. y-intercept

Find the y-intercept for each graph.

1. 2. 3.

4. 6.

y-inter = initial value = y-inter = y-inter =

Understanding Slope and Y-Intercept

B. Finding the Slope 1. How do we find the slope of a line graphically?

2. What other words/phrases represent slope?

3. What is the slope formula?

4. What letter do we use to represent slope?

Find the slope for each graph.

5. 7.

Slope = m = Slope = Slope =

Find the slope and y-intercept for each graph.

m = Slope = rate of change =

b = y-inter = initial value =

Understanding Slope and Y-Intercept

m =

Slope y –intercept y-inter =

m = rate of change = Slope

b = initial value =

Draw the line through the given points and find the slope.

7. A(1,2) and B(1,-4) 8. A(3,4) and B(-3,-5) 9. A(5,-4) and B(-5,4)

Writing and Equation of a Line Equation of a Line (for diagonal lines)

To write the equation of a line we need 2 pieces of information:

&

The is the point where the line crosses the

The standard equation of a line is

The m represents the

The b represents the

A. Find the Slope and y-intercept from an Equation

Find the slope or y-intercept of each line.

1. y = 3x + 2 2. y = -5x + 1 3. y = x – 7 4. y = 2x – 8

Slope = m = y –inter = b =

5. y = ½ x 6. y = -x + 6 7. y = 2x 8. y = 2x + 4

m = Slope = b = y-inter =

B. Writing an Equation Given the Slope and y-intercept Write the equation of each line given (for diagonal lines):

1. Slope = 4 2. Slope = 1 3. Slope = 1/3 4. Slope = -3

y-inter = -3 y –inter = 8 y –inter = -2 y –inter = 5

5. Slope = -7 6. Slope = ½ 7. Slope= 2

8. Slope = - 1

3 5

y-inter = 0 y –inter = -3 y –inter = -6 y –inter = 6

Writing an Equation of a Line

C. Writing an Equation from a Graph What 2 pieces of information do we need in order to write the equation of a line?

&

Slope =

y-inter =

Equation

m =

b =

Equation

Slope =

y-inter =

Equation

D. Vertical and Horizontal Lines 1a. Name 3 points on the given line. 2a .Name 3 points on the given line.

1b. What do you think the equation is? 2b. What do you think the equation is?

Write the equation for each line.

3. 4. 5 6

Writing an Equation of a Line

1. Given y = -3x + 5, what is the slope of the line?

2. Given y = x – 4, what is the rate of change?

3. If the slope of a line is ½ and the y-intercepts is 3, what is the equation of the line?

4. What is the slope of a line whose equation is y = ½x - 2?

5. What is the slope and y-intercept of a line whose equation is y = 1

x 4? 3

Write the equation of the line given:

6. m = -6 and b = 2 7. Rate of change =

3

and initial value = 4 5

8. m = 3 and b = -4 9. slope = 2 and y-intercept = 0

Find the slope algebraically Slope

=

Rise

Run = ∆x

∆y

10. (0,0) and (3,-3) 11.

Write the equation of each line

12 13 14 15

x 4 8 12

y 8 6 4

Writing the Equation of a Line

1) Write the equation of the line if it passes through the point (1, 9) and has a slope of 2.

2) Write the equation of the line if it passes through the point (4, -1) and has a slope of -3.

3) Write the equation of the line if it passes through the point (-4, -5) and has a slope of ¾.

4) Write the equation of the line whose slope is -2 and passes through the point (6, -20)

5) Write the equation of the line whose slope is 2

and passes through the point (3,4) 3

6) Write the equation of the line if it passes through the point (-4, 2) and has a slope of ½ .

Writing an Equation of a Line Write the equation of the line for

each:

1) A line that passes through the point (-7, 10) and has a slope of -4

2) A line that passes through the point (6, 2) and has a slope of 2.

3

3) A line that passes through the point (6, 2) and has a slope of − 1

3

4) A line that passes through the point (-1, -4) and has a slope of 1.

5) Is the point (3, -4) on the line y = 3x – 6? Justify your answer

6) What is the formula for slope?

7) Simplify (-4x4y7)2 8) Sketch a line with a negative slope.

Writing an Equation of a Line

Now we are going to write the equation of a line given TWO POINTS!

What information are we missing in order to write the equation of a line?

How do we find the slope given 2 points?

Steps:

1)

2)

3)

Write the equation of the line for each:

1) A line that passes through the points (3, -6) and (-1, 2)

2) A line that passes through the points (4, -4) and (8, -10)

3) A line that passes through the points (3, 4) and (5, -4)

4) A line that passes through the points (-3, 1) and (-2, -1)

Writing an Equation of a Line

1) A line that passes through the points (1, 2) and (3, 4)

2) A line that passes through the points (2, -2) and (4, -1)

3) A line that passes through the points (2, -4) and (6, -2)

4) A line that passes through the points (0, 3) and (2,0)

5. Given the points (3,5), (4,7), (9, 13), find the domain.

6. Given the equation y = x + 5, what is the slope?

7. If the initial value of a line is 3 and the rate of change is -2, what is the equation of the line?

8. What is the equation of the line below

Solving an Equation for y

Solve each equation for y and state the slope and y-intercept.

1. 2x + y = 5 2. –x + 2y = 12 3. 3x – 4y = 8 4. x – y = 7

5. 3x = y + 1 6. y – 4x = 4 7. 5x + 5y = 15 8. 9y – 2x = 27

9) Evaluate each expression for n = 3

a. 2n + 5 – n b. 3n+18

3n

c. 24

· n 4–n

10) Simplify:

a) 54∙58 b) 4x2(3x) c) (5x)0 d) 5x0 e) 6-4∙6-3

Graphing a line from a table

A) Steps for Graphing a Linear Equation from a table

1.

2.

3.

4.

1. y = 2x - 5 2. y = 1

x + 2 3

3. 2x + y = 1 4. 4y + 2x = 16

x y (x,y)

x y (x,y)

x y (x,y)

x y (x,y)

Graphing a line from a table

x y (x,y)

3. 6 + y = 2x 4. 2x – y = 4

x y (x,y)

x y (x,y)

x y (x,y)

Try These: 1. y = 3x - 4

2. y = -x

Graphing a line from a table

Graph each line using a table of values.

1. y = 3x - 6 2. y =

1 x +1

2

3. -3x + y = -4 4. 3y + 6x = 9

x y (x,y)

x y (x,y)

x y (x,y)

x y (x,y)

Graphing a Linear Equation using Slope and Y-intercept

Graphing Linear Equations without a table

Graph the line: y = 2x - 5

Steps

1)

2)

3)

4)

5)

Graphing a Linear Equation using Slope and Y-intercept

1) y

x

2) y x

3) y x

4) y

x

5) y x 6) y

y

x

y

x

y

x

y

x

y

x

y

x

Graphing a Linear Equation from the Slope and Y-Intercept

Graph the following lines.

1) y = -2x + 1 2) y = 3x

3) slope = 0 y-intercept = -3 4) x = 2 5) y = -5

6) Given y = 5 - 3x , what is the slope of the line?

7) Given y = -7 - 4x, what is the y -intercept of the line?

8) If the slope of a line is ½ and the y-intercepts is 3, what is the equation of the line?

Graphing a Linear Equation from the Slope and Y-Intercept

Sketch the graph of each line.

1) x y

2) x y

3) x y 4) x

y

x

y

x

y

x

y

x

Graphing a Linear Equation from the Slope and Y-Intercept

5) x y 6) x y

7) x y 8) x y

9) y 10) x y

y

x

y

x

y

x

y

x

y

x

y

x

1) y = -5x 2) y = -5

4s

Graphing a Linear Equation from the Slope and Y-Intercept Graph the following using slope/y-intercept: Name the type of slope for each line.

2

3) y = - 3

x + 7 4) 2x + y = 4

5) Simplify each expression:

5 5 3

3 2 –8 46∙65∙43

a) 2s2 b) (2x ) c) 7 ∙ 5 ∙ 7 d)

4∙62

6) Rewrite 81 in exponential form using 3 as the base. 7) Solve: 8 + 1

x = −1 + 3

x 2 4

Function Rules

Vocabulary:

Input values:

Output values:

Relation:

Function:

Function Rule:

Making Function Tables

To find the output values of a function, substitute the input values for the variable in the function rule.

1) y = 2x + 1

2) y = x + 2 3) y = x – 4

4) What is the output for an input of 7 if the function rule is 4n?

5) If the output is 4 and the function rule is n + 3, what is the input?

Input (x)

Output (y)

-1

0

2

Input (x)

Output (y)

2

4

8

Input Function Rule Output Ordered pairs

x 2x + 1 Y (x,y)

0

1

2

Finding Function Rules

This year, the only function rules you will write will be linear equations. To write a function rule, then, is to

write a liner equation!

What is the slope?

What is the y-intercept?

Write the equation for the line (function rule)

Write an equation for each given function (Function rule).

1) 2)

x y

-2 -7

-1 -4

0 -1

3) 4)

x 6 4 2

y 3 2 1

Hours 20 25 35

Pay ($) 160 200 240

Input (x)

Output (y)

1 2

2 5

3 8

n t

1 4

2 8

3 12

Finding Function Rules Write the function rule for the following table, fill in the missing y-value in the table, and graph the function.

Input Output

-2 -4

-1 -2

0 0

1

Function Rule:

Write an equation for the function and find the missing value in the table:

2) 3)

m c

1 0

2 1

3 2

4 3

5 4

100

4) What is the output for the function rule y = -3x – 2 if the input is 10?

5) What is the input for the function y = 2x – 5 if the output is -11?

x y 0 0

1 20

2 40

3 60

4 80

27

x

5

6

7

8

y

4

3

2

x

3

8

5

8

y

4

3

2 0

x

2 3

y

3

4

5 6

Function vs. Non-Function

Vocabulary:

Relation:

Domain:

Range:

Function:

Vertical Line Test:

Given the relation: {(1,2), (2, 4), (3, 5), (2,6), (1,-3)}

What is the domain?

What is the range?

Complete the following table and graph the function:

x y

Which relation represents a function?

1) 2) 3) 4)

Which relation diagram represents a function?

5) 6) 7) 8) Domain

Sue

Joe

Emma

Lilly

Range

Blue

Red

Pink

x y

0 -4

1 -1

2 2

3 5

4 8

x y

0 1

2 1

4 1

6 1

8 1

x y

0 5

1 6

2 7

1 8

0 9

x y

12 -2

10 -1

8 0

10 1

6 2

12)

Domain

A B C D

Range

1 2

Domain

Beth Sally Lucy Jen

Range

Dave Mike Ryan Dan

Using the vertical line test state whether or not each relation is a function.

9) 10) 11)

Try These: Which of the following represents a function?

1) 2) 3) 4)

5) 6) 7) 8)

9) 10) 11) 12)

13) 14)

Which set of ordered pairs represents a

function?

1)

2)

3)

Which set of ordered pairs is not a function?

1)

2)

3)

x y

1 7

5 5

9 3

1 1

Domain

2 3 4 5

Range

1 2 3

13)

x y

5 2

7 3

9 4

11 5

5 6

x y

-2 2

-1 5

0 4

1 5

Function vs. Non-Function

1) Which of the relations below is a function?

A) {(2,3), (3,4), (5,1), (6,2), (2,4)}

B) {(2,3), (3,4), (5,1), (6,2), (7,3)}

C) {(2,3), (3,4), (5,1), (6,2), (3,3)}

2) Given the relation A = {(5,2), (7,4), (9,10), (x, 5)}. Which of the following values for x will make relation A a function?

A) 7 B) 9 C) 4

3) The following relation is a function.

{(10,12), (5,3), (15, 10), (5,6), (1,0)}

A) True B) False

4) Which of the relations below is a function?

A) {(1,1), (2,1), (3,1), (4,1), (5,1)}

B) {(2,1), (2,2), (2,3), (2,4), (2,5)}

C) {(0,2), (0,3), (0,4), (0,5), (0,6)}

5) The graph of a relation is shown at the right. Is this relation a function?

A) Yes

B) No

C) Cannot be determined from a graph

6) Is the relation depicted in the chart below a function?

A) Yes

B) No

C) Cannot be determined from a chart

7) The graph of a relation is shown at the right. Is the relation is a function?

A) Yes

B) No

C) Cannot be determined from a graph

Linear vs. Non-Linear

Are the following graphs Linear or Non-linear? (Which ones are Linear Functions?)

7)

Are the following equations Linear or Non-linear? (Which ones are Linear Functions?)

9) y x 3 3x 9 10) y x 2 11) y

2 10

x 12) y x

2 x 2

13) y = 5x 14) y = 2 15) 16) x = 8 16) y = |x + 7|

Are the following tables Linear or Non-linear? (Which ones are Linear Functions?)

17) 18) 19) 20)

Are the following word problems Linear or Non-linear?

21) Sam put $10 in the box under his bed every week

22) A dolphin jumps above the surface of the ocean water, then dives back in the water.

23) A soccer player sprints from one side of the field to the other.

24) A lacrosse player throws a ball upward from her playing stick with an initial height of 7ft and an initial velocity of 90 ft. per second.

25) A rocket is shot off into the air and then comes back down to the ground.

3) 4)

5) 6) * 8)

3

Linear vs. Non-Linear

Are the following Linear or Non-linear?

1) y = x2 − x − 2 2) y = |x + 1| 3) y = 5x + 2 4) y = x3 − 3x + 9 5) y − 7x = −2

6) 7) 8) 9)

10) A baseball player hits a pop fly

11) The path traveled by a basketball during a shot on the basket

12) A babysitter getting paid $6 per hour

13) You deposit $250 per year for 39 years

14) 15) 16) 17)

18) Which equation represents a linear function?

A. y = 8x4

B. y = 0.05x – 0.01

C. y = 2x2

+ 5

D. √x

19) Which of the following does not describe a linear function? A. the perimeter, p, of a square with side s: p = 4s

B. the circumference, C, of a circle with radius r: C = 2nr C. the salary, s, of an employee making $12.50 per hour, h: s = 12.50h

D. the area, A, of a circle with radius r: A = nr2