Algebra 1: 4.1 Notes
NAME: _______________________________________
Identifying Linear Functions
Linear Function =
Example of Linear Function =
Example 1:
Identify whether each graph represents a function. Explain. If the graph does represent a function, is the function
linear?
Linear Function?
Example 2:
Identify whether each graph represents a function. Explain. If the graph does represent a function, is the function
linear?
Linear Function?
You Try 1:
Identify whether each graph represents a function. Explain. If the graph does represent a function, is the function
linear?
Linear Function?
You Try 2 :
Identify whether each graph represents a function. Explain. If the graph does represent a function, is the function
linear?
Linear Function?
Linear Function from a Table
Two things must be true to have a linear function in a table:
1)
2)
X
Y
-2
7
-1
4
0
1
1
-2
2
-5
Example 3:
Tell whether each set of ordered pairs satisfies a linear function. Explain.
X
Y
2
4
X
-10
Y
10
5
3
-5
4
8
2
0
2
11
1
5
0
You Try 3:
Tell whether each set of ordered pairs satisfies a linear function. Explain.
X
X
Y
Y
3
5
-4
13
5
4
-2
1
7
3
0
-3
9
2
2
1
11
1
4
13
Standard Form: Ax  By  C
Linear
Not Linear
3x + 2y = 10
3xy + x = 1
y–2=x
x3 + y = -1
-y = 5x
x + (6/y) = 12
1)
2)
3)
Example 4:
Tell whether each function is linear. If so, graph the function
y=x+3
X
Y
Work
Example 5:
Tell whether each function is linear. If so, graph the function
y = 12
X
Y
Work
You Try 4:
Tell whether each function is linear. If so, graph the function
y = 5x – 9
X
Y
Work
You Try 5:
Tell whether each function is linear. If so, graph the function
y = 2x
X
Y
Work
Application 1
Sue rents a manicure station in a salon and pays the salon owner $5.50 for each manicure she gives. The amount Sue
pays each day is given by f(x) = 5.50x, where x is the number of manicures. Graph this function and give its domain and
range.
X
Y = f(x)
Work
0
1
2
3
4
Why did I choose only positive x-values for this problem?
Domain(the possible x-values that make sense in this problem):
Range(the possible y-values that make sense in this problem):
Application 2
At another salon, Sue can rent a station for $10.00 per day plus $3.00 per manicure. The amount she would pay each
day is given by f(x) = 3x + 10, where x is the number of manicures. Graph this function and give its domain and range.
X
Y = f(x)
Work
0
1
2
3
4
Domain:
Range:
4.2 Note-Sheet: Graphing Using Intercepts
How to graph using intercepts
x – intercept
x – intercept:
(#, 0)
**NOTE:
TO FIND:
Plug _____ in for ______ and solve for _____
Example 1: 3x – 2y = 12
You Try 1: -3x + 5y = 30
The x- intercept is __________
The x- intercept is __________
y – intercept
y – intercept:
(0, #)
**NOTE:
TO FIND:
Plug _____ in for ______ and solve for _____
Example 2: 3x – 2y = 12
You Try 2: 4x + 2y = 16
The y- intercept is __________
The y- intercept is __________
Now let’s put it all together to graph . . .
STEPS
1) Find the x – intercept
2) Find the y – intercept
3) Plot the two points
4) Connect Points with a straight line
Example 3: 2x – 4y = 8
Work for
x-intercept
Work for
y-intercept
Example 4:
Work for
x-intercept
2
1
y4 x
3
2
Work for
y-intercept
x-intercept = ______________
y-intercept = ______________
x-intercept = ______________
y-intercept = ______________
You Try 3: -3x + 4y = -12
Work for
x-intercept
Work for
y-intercept
x-intercept = ______________
y-intercept = ______________
Algebra 1:
NAME: _________________________________
4.3 Note Sheet
Slope is ___________________________________________________________________________________________
Types of Slope
Positive
Negative
Mountain Picture
Slope (m) =
=
Finding Slope Given a Graph
1)
2)
3)
4)
Example 1:
Positive, negative, 0, or undefined slope?
Rise:
Slope:
Run:
Zero (0)
Undefined or No Slope
Example 2:
Positive, negative, 0, or undefined slope?
Rise:
Run:
Slope:
Example 3:
Positive, negative, 0, or undefined slope?
Rise:
Run:
Slope:
You Try!!!!!
Positive, negative, 0, or
undefined slope?
Rise:
Run:
Slope:
Positive, negative, 0, or
undefined slope?
Rise:
Run:
Slope:
Positive, negative, 0, or
Positive, negative, 0, or
undefined slope?
undefined slope?
Rise:
Rise:
Run:
Run:
Slope:
Slope:
4.4 Note-Sheet: Slope Formula
If given 2 points on a line, you may find the slope using the formula:
Any 2 points on the line: ( x1 , y1 )( x2 , y2 )
Slope Formula:
m = __________ = ________________
Example 1
Find the slope of the line that passes through the points (3, 4) and (-6, -2)
Example 2
Find the slope of the line that passes through the points (7, 10) and (3, 2)
You Try 1: Find the Slope
1)
(-2, -6) (3, 5)
2)
(-2, 4) (5, -6)
3)
(2, -3) (-2, 15)
Finding Slope Given a Table
Example 3
0
Pick ANY two points and write as an ordered pair (x, y)
0
Apply slope formula
X
1
2
3
4
5
6
7
8
9
Y
13
17
21
25
29
33
37
41
45
You Try 2: Find the Slope
0
Pick ANY two points and write as an ordered pair (x, y)
0
Apply slope formula
X
48
43
38
33
28
23
18
13
8
Y
32
38
44
50
56
62
68
74
80
Finding Slope From a Graph
0
Pick ANY two points and write as an ordered pair (x, y)
0
Make sure the point crosses at an intersection
0
Apply the slope formula
Example 4: Find the slope using the formula
You Try 3: Find the Slope
Applications of Slope
Slope is sometimes referred to as the “rate of change” between 2 points.
In real-world problems, finding the slope can give you information about how a quantity is changing.
Example 5
Example 6
Find the slope. Then tell what the slope represents
represents
Find the slope. Then tell what the slope
You Try 4: Find the Slope
Find the slope. Then tell what the slope represents
4.6 Note-Sheet: Slope-Intercept Form
3 Parts of the Lesson:
1 – Identifying slope and y-intercept from an equation
2 – Graphing a slope-intercept equation
3 – Writing an equation in slope-intercept form from a graph
Slope-Intercept Form
y-intercept
(where the
graph
crosses the
y-axis)
Slope (m)
rise/run
Identifying Slope and y-intercept from an equation
1) Solve the equation for y.
2) Put into slope-intercept form (y = mx + b)
(HINT: the slope is always next to the “x”)
Example 1:
y = 4x – 3
You Try 1:
m = ________
m = ________
b = ________
b = ________
y
2
x
3
Example 2:
3x + 2y = 8
You Try 2:
m = ________
m = ________
b = ________
b = ________
6x + 2y = 10
How to Graph
STEPS
1) Solve for y so the equation is in slope-intercept form (y = mx + b)
2) Identify the slope and the y-intercept
m = _______ b = _______
3) Put a point on the graph where the y-intercept is
4) From the y-intercept, use the slope to plot at least two more points
5) Connect the points with a straight line
Example 3:
2y + 10 = 4x
m = _____
You Try 3:
y = 2x + 1
m = ______
b = _______
b = ______
How to Write an Equation from a Graph
STEPS
1) Find b (the y-intercept)
2) Find the slope (m) using any 2 points --- (rise/run)
(HINT: Don’t forget to ask yourself if the slope is positive, negative, 0, or undefined)
3) Put both values into the slope-intercept formula (y = mx + b)
Example 4:
m = ______ b = ______
y = ___x + ___ (y = mx + b)
You Try 4:
m = ______ b = ______
y = ___x + ___
Special Cases
Horizontal Lines
Vertical Lines
The equation: y = #
The equation: x = #
(y = because it crosses the y-axis)
(x = because it crosses the x – axis)
m = __________
m = __________
b = ___________
b = ________
Graphs of Special Cases
Write the equation of each special case below.
Algebra 1:
NAME: _______________________________________
4.7 Notes: Point-Slope Form
 Write each equation in slope-intercept form (y = mx + b) Solve for y.
 STEP 1: ______________________________________
 STEP 2: ______________________________________
y – 149 = 9(x – 16)
Point-Slope Form
y – 23 = 5(x – 4)
y + 20 = 7(x – 10)
y  y1  m( x  x1 )
NEEDED: ______________________________________________________________
Plug __________________________________________________________________
 Write an equation in slope-intercept form
x1 y 1
m = 3 and (2, -7)
y  y1  m( x  x1 )
(use the point-slope formula)
m = -4 and (4, 2)
m = -5 and (2, 3)
Writing an equation given 2 points
STEP 1:
STEP 2:
STEP 3:
 Given the following points, write an equation in point-slope form and slope-intercept form
(-1, 10) (5, 8)
(5, 65) (7, 71)
Write an equation given the graph, using point-slope form.
STEP 1:
STEP 2:
STEP 3:
(3, -6) (-5, 2)
Write an equation from the given information below.
1) Contains the origin and has slope 5
2) Crosses y-axis at 2 and has slope of -1
What Point is the origin? ________
What point crosses the y-axis at 2? ________
3) Contains the origin and point (2, 3)
4) Crosses the x-axis at x =2 and y-axis at y = 6
What point crosses the x-axis at 2? ________
What point crosses the y-axis at 6? ________
5) Crosses x –axis at x = -5 and y-axis at y = 0
What point crosses the x-axis at -5? ________
What point crosses the y-axis at 0? ________