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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