Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Unit 4:
Function Characteristics
Lesson 1 (PH Text p.60; 1.9): Graphing in the Coordinate Plane
Lesson 2 (PH Text 4.1): Using Graphs to Relate Two Quantities
Lesson 3 (PH Text 4.2 – 4.4): Patterns and Functions AND
Graphing Function Rules
Lesson 3a (PH Text p.260-261): Graphing with a Calculator
Lesson 4 (PH Text 4.5): Writing a Function Rule
Lesson 5 (PH Text 4.6): Formalizing Relations and Functions
Lesson 6 (PH Text 5.1): Slope and Rate of Change
Lesson 6a (PH Text p.305): Graphing Calculator Investigation
Lesson 7 (PH Text 5.3): Slope-Intercept Form y=mx+b
Lesson 8 (PH Text 5.4): Point-Slope Form y – y1 = m ( x – x1 )
Lesson 9 (PH Text 5.5): Standard Form Ax + By = C
Lesson 10 (PH Text 5.6): Parallel and Perpendicular Lines
Lesson 11 (PH Text 5.7): Scatter Plots and Trend Lines (Linear Regression)
Lesson 12 (PH Text 6.5): Graphing Linear Inequalities
1
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 1 (PH Text p.60; 1.9): Graphing in the Coordinate Plane
Objective:
to use tables, equations, and graphs to describe relationships
Describe and Label:
Coordinate Plane –
x-axis –
y-axis –
origin –
Quadrant I –
Quadrant II –
Quadrant III –
Quadrant IV –
ordered pair – example: (3, -2)
x-coordinate –
y-coordinate –
Solution of an equation –
Is (5, 20) a solution to the equation y = 2x?
(2, 4)?
(3, 6)?
(5, ___)?
There are three primary ways to represent the relationship between two variables.
TABLE
EQUATION
GRAPH
x
y
2
3
5
When graphing the relationship from the table or equation, use the x- and y-values to
create ordered pairs. Plot those points on the graph. Connect the points.
2
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Inductive Reasoning – using observable patterns to reach a conclusion
Example:
Freddy Hill is selling fruit baskets. The numbers of oranges compared to the numbers of
apples in their most commonly sold baskets is represented on the graph below. Based on the
pattern observed in the graph, how many apples would you expect to find in a basket
containing 10 oranges?
(You used inductive reasoning to answer that question.)
If the pattern continues, how many oranges would be in a 30apple basket?
What is the pattern?
Can you describe the pattern using an equation?
(Hint – use x for oranges and y for apples)
Billy Hill sells fruit baskets with a different combination of fruit.
Using inductive reasoning, how many apples would you expect to
find in a basket containing 10 oranges?
How many oranges would be in a 30-apple basket?
What is the pattern?
Can you describe the pattern using an equation?
HW: p.64 #12-18, 22-25, 36-37, 42-45; graph below #1-10
(need coord. grids for #17-18, 22, 25)
3
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 2 (PH Text 4.1): Using Graphs to Relate Two Quantities
Objective: to represent mathematical relationships using graphs
4
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Look at and analyze the variety of graphs. Explain how the graph helps us understand the relationship
between the two variables.
HW: p.237 #1, 3, 4, 8-13, 15, 17-23
5
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 3 (PH Text 4.2 – 4.4): Patterns and Functions AND Graphing Function Rules
Objectives: to identify and represent patterns that describe linear functions
to graph equations that represent functions
Dependent Variable –
Independent Variable –
Inputs –
Outputs –
Function – a relationship that pairs each input value with exactly one output value –
Linear Function –
Non-Linear Function –
Function Rule – a variable expression that describes a function (an equation or inequality)
Function Table – a table that shows the x and y values for a function
Examples:
x y
x y
y=x–4
y = 2x
x x–4
x 2x
8 4
-3
6 2
-2
4 0
0
2 -2
1
0 -4
4
Class Practice:
Determine if each relation is a function by looking at the x and y values. (Is there only one y for each x?)
x
3
5
-7
15
y
2
-11
13
4
x
4
7
-12
7
y
3
-9
10
4
x
3
5
-7
15
6
y
2
2
2
2
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Functions can be modeled by: tables, rules (equations) and/or graphs.
TABLE
x
x
4
2
0
-2
-4
y
x+3
7
5
Additional Class Practice:
y=x–4
x y
x x–4
8 4
6 2
4
2
0
EQUATION
GRAPH
y=x+3
f (x) = x + 3
x
x
-3
-2
0
1
4
y
2x
y = 2x
Reminder:
continuous data –
discrete data –
HW: p.243 #2, 5, 13-16; p.257 #8-9, 20-23, 33-36, 44-45 (need coord. grids for #13, 15, 16, 9)
7
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 3a (PH Text p.260-261): Graphing with a Calculator
Objectives: to graph functions using a graphing calculator
Notes:
Exercises #1-6:
7. Xmin ____,
Xmax ____,
Ymin ____,
Ymax ____
8. 2x + 3y = 6 on a calculator?
HW: Practice 4-4 (two pages - without a calculator)
8
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
x
y
x
x
y
9
y
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
11.
Writing Describe the general shape of the function y = |x|.
Extra coordinate grids:
10
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 4 (PH Text 4.5): Writing a Function Rule
Objectives: To write equations that represent functions
A function rule is often useful to describe real life situations. Solving the function can then help us
find a solution to a problem.
Example:
While visiting NYC, we wanted to take a taxi to Times Square. The taxi charged an initial rate of $1.50, plus
$0.15 per minute. Write a function rule to describe the cost for the ride. If we estimate that it will be a 20minute ride, what would our estimated cost be?
f (x) =
f (20) =
Class Practice:
1) Write a function rule to describe the cost to buy apples at $1.29 per pound.
2) Write a rule to calculate the cost to rent a yurt if the state park charges $50 per yurt, plus $10 per camper.
3) Write a function rule to describe the change you would receive when making a purchase using a $20 bill.
Additional Example:
f (x) = 1.5 x
or
Write a function rule to describe the cost to buy grapes at $1.50 per pound.
y = 1.5 x
purchase
x
3
5
1.5
2
2.25
4
HW: p.264 #6-16, 33-36, 43-45
11
change
f(x)
$4.50
$7.50
$2.25
$3.00
$3.38
$6.00
What do the different parts
of the graph mean?
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 5 (PH Text 4.6): Formalizing Relations and Functions
Objectives: To determine whether a relation is a function
To find the domain and range of a relation and use function notation
Relation- is a set of one or more ordered pairs
Domain- is the set of all first elements or x-coordinates ( x- values ) – the set of all possible input values
Range- is the set of all second elements or y-coordinates ( y- values ) – the set of all possible output values
Functions are often described using a domain and a range.
Example:
p  4s
p = 4(2)
p = 4(5)
p = 4(6)
p = 4(12)
If the domain = {2, 5, 6, 12}, what is the range?
p=
p=
p=
p=
range = {___, ___, ___, ___}
Is the relation a function?
3a.
5
6
8
7
9
6
7
Examples:
1) State the relation specified in the table below as a set of ordered pairs. State the domain and range of
this relation.
x
y
-1
0
0
1.5
1
3
2) Given the domain {-2, 0, 1} of the relation 2x + y = -1,
determine the range. Graph the results.
12
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Additional ways to write a function rule:
Arrow Notation – shows the number pairs – you write, “x
x – 4” ; you say, “x is paired with x – 4”
Function Notation – a method used to state a function rule, similar to y = x – 4
You write, “f (x) = x – 4”; you say, “f of x equals x – 4” … This shows that f (x) is a function of x.
Function notation is useful in identifying the value of a function at a certain value of x.
For example, for f (x) = x – 4, f (9) = 5.
Examples:
1) Evaluate, given f(x) = x – 3 and g(x) = x 2
a) f(0)
b) g(2)
e) f(4) + g(-2)
f) g(-2) + f(6)
c) f(0) – g(2)
d) g(5) – f(1)
g) f(4) – g(-2)
h) g(-2) – f(6)
2) Evaluate, given the domain {-2, 0, 1}
a) y = 4x – 1
b) g ( x) 
13
1 2
x 2
3
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Function- is a relation in which each element of the domain is paired with exactly one element of the range.
(exactly one y for each x)
Example
A
Example
B
Vertical Line Test - a method used to determine if a relation is a
function or not – if a vertical line passes through a graph more than
once, the graph is not the graph of a function
Example A: The vertical line passes through the graph at only one
point at a time, so this IS the graph of a function.
Example B: The vertical line passes through the graph at more than
one point at a time, so this IS NOT the graph of a function.
Draw some possible examples of graphs that are functions.
Draw some possible examples of graphs that are NOT functions.
Do together?: p.271 #1-8
HW: p.271 #1-8, 12-18, 21, 24, 30-34 even, 37-44; problems below
Evaluate, given that f(x) = 3x – 1 and g(x) = x2 + 2.
1. f(2) ______
2. f(-3)
5. f(2) + g(2) ______
______
3. f(-6)
6. f(-3) + g(0) ______
______
4. g(2) ______
7. f(-2) · g(-2) ______
8. g(-2·4) _______
4.1-6 Review
o p.282 #5-24
o p.285 #1-12, 17-18
After ch.4 quiz HW: p.289 all
14
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 6 (PH Text 5.1): Slope and Rate of Change
Objective: to find the rate of change/slope of a line from its table or graph, or from the coordinates of two
points of the line
Ordered pairs can be numbered: (x2, y2) and (x1, y1) The sub-numbers just identify them as different points.
Slope of a line is
– the ratio of the change in y to the change in x between any two points on a line.
– the measure of the steepness of a line
– the ratio of the vertical change to the horizontal change
A slope will have a predictable “look”.
Positive Slope –
Negative Slope –
Zero Slope –
Undefined Slope –
Practice:
Find the slope of each line.
3.
4.
5.
7.
Explain why the slope of a
vertical line is always
undefined.
8.
6.
Explain why the slope of a
horizontal line is always zero (0).
Given the ordered pairs (x2, y2) and (x1, y1), the following is true:
Slope
= ∆y =
∆x
vertical change
horizontal change
=
Rise
Run
=
y2 – y1
x2 – x1
Find the slope of :
a line that contains the points M(12, 4) and N(6, 2)
a horizontal line that contains the points M(0, 2) and N(5, 2)
a vertical line that contains the points P(-3, 4) and R(-3, 2)
15
where x2 – x1 ≠ 0
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Find the slope of a line
through ( –2, 1) and (3, 5).
Then graph it to check.
Find the slope of a line
through ( 4, 0) and (–1, 5).
Then graph it to check.
Draw a line through the point (1, 2)
with the slope -3/2.
Draw a line through the point (–1, –3)
with the slope 5/4.
Collinear points - are points that lie on the same line.
If two line segments share a point and have the same slope, they are collinear.
Determine if points A(-4, -1), B(-2, 1), and C(2, 5) are collinear.
Since we can’t be 100% sure from the graph, we use the slope to help.
16
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Rate of change - allows you to see the relationship between two quantities that are changing
Of the two quantities, one is independent (x) and one dependent (y).
Rate of Change:
change in the dependent variable
change in the independent variable
(looks like slope)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
You often use the rate of change to find the amount of one quantity per one unit of another - so many
times your goal is to get a 1 (one) in the denominator, like “unit rate”.
Example:
Find the rate of change in the following problems and graph your results.
1. The snowy cricket chirps 100 times per minute at 65 degrees, and 128 times per minute at 72 degrees.
What is the rate of change? Explain what this means.
Rate of Change:
change in the dependent variable =
change in the independent variable
# cricket chirps =
degrees
128 – 100
72 – 65
= 28 = 4
7
1
The rate of change is ________, which means that the crickets chirp ____ more times per minute for
every degree the temperature increases.
17
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Reminder: A Linear Function is a function whose graph is a line. In a linear function, the rate of change is constant. When
multiple linear functions are graphed on the same coordinate plane, the line with the steeper slope (therefore the greater slope)
would indicate a greater rate of change.
2. A car rental company charges customers $30 for the first 100 miles and an additional fee for every mile
over 100 miles they drive. Using the table below, find the rate of change charged for the extra mileage.
Miles
100
150
200
250
Fee
$30.00
$42.50
$55.00
$67.50
3. Laura wished to measure the effectiveness of her new exercise program. At the end of every week she
counts the number of sit-ups that she can do in one minute. If she does 30 sit-ups the first week, 42 situps the second week, and 54 sit-ups the third week, is her progress linear?
Practice:
3
22. Describe how to draw a line that passes through the origin and has a slope of .
5
Each pair of points lies on a line with the given slope. Find x or y.
23. (2, 2), (5, y); slope = 2
24. (9, 4), (x, 6); slope = -1/3
HW: p.295 #7-19, 23, 24-38 even, 60
18
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 6a (PH Text p.305): Graphing Calculator Investigation
Objective: to investigate y = mx + b
A “parent function” is the simplest function with these characteristics. Graph the linear parent function y = x.
Using the arrow keys, arrow to the line icon on the left of the y. Make the line for this function bold by using
the enter key to toggle between options. Once the line icon is bold, arrow back to the right of the =. Leave
the parent function as Y1 as you test all the other lines using Y2.
Make observations of how the different numbers impact the graph.
19
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Match each equation with its graph; then check with the graphing calculator.
20
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 7 (PH Text 5.3): Slope-Intercept Form y=mx+b
Objective:
to write linear equations using slope intercept-form
to graph linear equations in slope intercept-form
Parent Function – the simplest function with certain characteristics
Linear Parent Function – the simplest linear function
=>
y = x or f(x) = x
Linear Equation – an equation that models a linear function – the highest power of x in a linear function is 1
y-intercept – the y-coordinate of a point where the graph crosses the y-axis
Slope-Intercept Form of an equation => y = mx + b
where: m = the slope of the line
b = the y-intercept
x represents the independent variable
y represents the dependent variable
What are the slope and y-intercept of the line for each equation?
7
3
1.
y = 3x – 5
2.
y  x
6
4
3.
4
y x
5
m = _____
m = _____
m = _____
b = _____
b = _____
b = _____
Write the equation of a line with the given slope and y-intercept.
2
1
4.
5.
m  , b = -5
m , b=0
3
2
6.
m = 0, b = -2
y = ___ x + ___
To test the coordinates of a point to see if the point lies on a particular line, substitute the x- and ycoordinates to see if they make the equation true.
Does the point (2, 4) lie on the line with the equation y = 3x – 6 ?
Does the point (3, 3) lie on the line with the equation y = 3x – 6 ?
21
____ = 3(____) – 6
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
To graph a linear equation: ① plot the y-intercept and ② use the slope to count to plot another point.
③ Draw your line. Then check for other points using the slope to verify them.
Graph the equation y = 3x – 1
Graph the equation
3
y  x2
2
1)
y-intercept =
(plot point)
1)
y-intercept =
(plot point)
2)
slope =
(count and plot points)
2)
slope =
(count and plot points)
3)
draw line
3)
draw line
__________________________________
Rewrite an equation if it is not in slope-intercept form.
the line.
Rewrite -5 = 4x – y so it is in y = mx +b form.
Then graph
__________________________________
To write an equation for a line from a graph, get the slope and y-intercept from the graph and substitute
into the form y = mx +b.
1. Find the y-intercept.
b = _____
2. Find the slope using two points.
m = _____
3. Substitute into y = mx + b to write the equation for the line.
22
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Find the equation of the line that passes through ( 1, 4 ) and ( 3, 10 ).
y2 – y1
x2 – x1
= _
–
–
1.
Find the slope.
2.
Use the slope and the coordinates of one point to find b by plugging into y = mx + b.
____ = ____ (____) + b
= ___
m=
Substitute known values for m, x, and y.
Solve for b.
3.
Substitute the slope and y-intercept into the slope-intercept form.
HW: p.310 #12-30 even, 40-60 even (48 is optional)
23
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
24
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 8 (PH Text 5.4): Point-Slope Form y – y1 = m ( x – x1 )
Objective:
to write and graph linear equations using point-slope form
Point-Slope Form of an equation => y – y1 = m ( x – x1 )
derived from the formula for slope =>
Point - Slope Form
y2 – y1
x2 – x1
= m
y – y1 = m ( x – x 1 )
m = slope
( x1, y1 ) is a given point
Find the equation of the line that has a slope of ½ and passes through the point (-4, -5).
y – ____ = ____ ( x – ____)
Substitute known values for m, x, and y.
Practice 1)
Write an equation of a line that has a slope of -4/5 and contains P (1, -2).
Practice 2)
Write an equation of a line that has a slope of 12 and contains Q (-¾, 8).
Graph the equation y – 2 = -2/3 ( x + 5 ).
To graph using the point-slope form, …
1.
Find the coordinates of the point. ( ____ , ____ )
2.
Plot the point.
3.
Find the slope.
4.
Count and plot points.
Practice 3)
Graph the equation y + 1 = ¼ ( x – 6 ).
Look at the graph. Find the needed information to write the equation.
25
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Example:
Write an equation in slope-intercept form of the line that passes through points ( 1, 4 ) and ( 3, 10 ).
y2 – y1
x2 – x1
= _
–
–
1.
Find the slope.
= ___
2.
Use the slope and the coordinates of the first point in point-slope form.
y – ____ = ____ ( x – ____)
m=
Substitute known values for m, x, and y.
3. Rearrange terms in order to write the equation of the line in the form slope-intercept form (y = mx + b).
Double check … Try this again using the second point and see if the equation is the same.
Practice 3)
Find the equation of the line that passes through (–2, 1 ) and ( 3, 4 ).
Use a table to write an equation.
The table to the right logs the gas used by Zander on his cross-country trip.
Write an equation in slope-intercept form to represent his gas usage.
1.
Use two points to find the slope.
2.
Substitute the computed slope, and
a data point into point-slope form.
3.
Rewrite.
HW: p.316 #6-24 even, 27, 34-36
Zander's
Gasoline Usage
Miles
200
475
760
1,010
1,300
1,600
1,875
2,200
2,450
(need 3 sm. coord. grids)
Good Idea: Mid-chapter Quiz – p.319 #1-6, 10-26
26
Gallons
of Gas
5,200
12,350
19,760
26,260
33,800
41,600
48,750
57,200
63,700
Algebra 1
Mrs. Bondi
Unit 7 Notes: Graphing Linear Functions
Lesson 9 (PH Text 5.5): Standard Form Ax + By = C
Objective:
to graph linear equations using intercepts
to write and linear equations in standard form
x-intercept – the x-coordinate of a point where the graph crosses the x-axis
y-intercept – the y-coordinate of a point where the graph crosses the y-axis
Standard Form of an equation => Ax + By = C
where: A and B are integers, and A and B are not both zero;
x and y represent the independent and dependent variables
Find the x- and y-intercepts of the graph of 3x + 4y = 12.
To find the x-intercept, substitute 0 for y.
Solve for x.
To find the y-intercept, substitute 0 for x.
Solve for y.
The x-intercept is ( ___ , 0).
The y-intercept is ( 0 , ___).
Graph 3x + 4y = 12.
Graph -3x – 4y = 12.
1. Find and graph the x-intercept.
2. Find and graph the y-intercept.
3. Draw the line containing the
intercepts.
Horizontal and Vertical Lines: Write the equation for each line in Standard Form.
Graph y = 2.
Graph x = -3.
Notice: NO x-intercept
Exception:
Notice: NO y-intercept
Exception:
27
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
NOTE: In Standard Form, A, B, and C must be integers. Ax + By = C
Write an equation in Standard Form for the line with the given point and slope.
1)
( –1, 7 ) and m = -1/2
2)
( 3, –35 ) and m = 5/3
3)
(2 , –7) and m = 2/3
4)
(9 , 7) and m = –4
5)
(–1 , 2) and m = -4/5
6)
(0 , 0) and m = 6
Look at the graph. Find the needed information to write the equation.
Write an equation to model a purchase decision.
You have a $25 iBuy gift card for electronic media. Write a model using the standard form of a linear
equation to represent your possible purchases that include games at $2 each and movies at $8 each.
28
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Linear Equation Summary:
You can describe any line using one or more of these forms of a linear equation. Any two equations
for the same line are equivalent.
Standard Form of an equation =>
Ax + By = C
Slope-Intercept Form of an equation => y = mx + b
Point-Slope Form of an equation =>
y – y1 = m ( x – x1 )
You can graph the line from any equation form by finding coordinates of points and/or the slope of the
line in the equation.
Practice: Use the information given to write an equation for the line in each form.
1.
2.
point: (-1, 2); m = -1
Slope-Intercept Form _______________________
points: (5, -1) & (-3, 4)
HW: p.323 #6, 7, 8-40 even, 43, 66-69
Point-Slope Form
_______________________
Standard Form
_______________________
Slope-Intercept Form _______________________
Point-Slope Form
_______________________
Standard Form
_______________________
(need 7 sm. coord. grids)
29
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
30
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
31
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Lesson 10 (PH Text 5.6): Parallel and Perpendicular Lines
Objective:
to use slope to determine if lines are parallel, perpendicular, or neither
to write equations of parallel or perpendicular lines
Parallel Lines (||) – two lines in the same plane that never intersect.
Perpendicular Lines (⊥) – two lines in the same plane that intersect at right angles.
Discovery: Graph the following lines from the given points.
Line 1
( –3, –2 ) and ( 1, 4 )
Line 1
( 1, 4 ) and ( 3, –2 )
Line 2
( 0, –6 ) and ( 4, 0 )
Line 2
( –1, 1 ) and ( 5, 3 )
m1 =
m1 =
m2 =
m2 =
These lines are ________________________.
These lines are ______________________.
Note:
Parallel Lines (||) have slopes that are ____________. (y-intercepts must be different)
Perpendicular Lines (⊥) have slopes that are _________________________________.
Find the slope of a line parallel to the graph of each equation.
6x + 2y = 4
y–3=0
-5x + 5y = 4
32
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Write an equation for a line that contains ( –2, 3 ) and is parallel to the graph of 5x – 2y = 8.
Reciprocal Reminder – A reciprocal (also called multiplicative inverse) is the number which when
a
b
multiplied by the original gives the multiplicative identity, 1. The reciprocal of a fraction
is .
b
a
Reciprocal Practice:
3
original fraction:
5
opposite reciprocal:
original fraction: – 8
opposite reciprocal:
1
2
opposite reciprocal:
original fraction: 
Find the slope of a line perpendicular to the graph of each equation
y = 3x - 2
y=x
3x + 5y = 7
Write an equation for a line that contains ( 0, 4 ) and is perpendicular to y = 2x – 2 .
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Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Classify Lines:
Determine if the graphs of y = ¾x + 7 and 4x – 3y = 9 are parallel, perpendicular, or neither. Justify your
response.
Determine if the graphs of 6y = -x + 6 and y = -1/6 x + 6are parallel, perpendicular, or neither. Justify
your response.
Math in the real world:
HW: p.330 #4-32 even, 38-40
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Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Lesson 11 (PH Text 5.7): Scatter Plots and Trend Lines (Linear Regression)
Objective:
to write an equation of a trend line and of a line of best fit
To use a trend line and a line of best fit to make predictions
to use a calculator to find the line of best fit in a scatterplot
Write an equation of a trend line.
A history teacher asked her students how many hours of sleep
they had the night before a test. The data below shows the
number of hours the student slept and their score on the exam.
Plot the data on a scatter plot. If there is a correlation, describe it.
Draw a trend line. Estimate the coordinates of two points on the
line. Use those points to write an equation.
The equation can be used to help us estimate an answer to a question, like the letter grade we might expect of
someone who slept 6 hours the night before the test.
Line of Best Fit – the most accurate trend line showing the relationship between two sets of data
– the easiest way to find this is with a graphing calculator
We estimate the line of best fit by finding the equation of a trend line.
Correlation Coefficient – r – tells how closely the equation models the data
– the nearer the r is to 1 or -1, the more closely the data cluster around the line of best fit
– the graphing calculator will give this
To use a graphing calculator to find the line of best fit, follow the directions provided.
The calculator will find the line of best fit by using something called a linear regression. It will show the line
of best fit in one of these two forms.
y = ax + b
or
y = a + bx
↑
↑
↑ ↑
slope y-intercept
y-intercept slope
Remember that slope is the coefficient of x in a linear equation!
You will then need to retrieve the information from the calculator and write your own equation – slopeintercept form is easiest.
35
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Practice:
Find the equation of the trend line; then use a calculator to find the line of best fit.
Equation of trend line:
Equation of the line of best fit:
Practice:
b. Equation of trend line:
c. Equation of the line of best fit:
Causation – when a change in one quantity causes a change in a second quantity. A correlation between
quantities does not always imply causation.
Examples of causation:
 The number of miles driven and the amount of gas used have a positive correlation AND a causal
relationship, because driving more miles uses more gas.
 The number of yearbooks delivered to Pennfield and the number of students in the school have a positive
relationship, but NOT a causal relationship. Having more yearbooks delivered to the school will not
cause there to be more students.
HW: p.337 #6-16 (need calc. for 11), 20 with calculator
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(need 5 coordinate grids)
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Lesson 12 (PH Text 6.5): Graphing Linear Inequalities
Objective: To graph linear inequalities and solve word problems.
Vocabulary:
Linear inequality – relates two linear expressions with an inequality sign
Solution of an inequality – any ordered pair that makes the inequality true
Half-planes – are the regions above and below the line formed from the inequality
Boundary – the line that splits the regions is the boundary of each half-plane
Graphing Inequalities:
Number Line Example
Coordinate Plane Example
x≤3
y
x≤3
x
A _________ _____________ describes a region of the coordinate plane that has a ____________ _____ .
Test a point on each side of the line to determine which side to shade. If the point works, that side of the line
is the solution region – so shade it.
Every point in the shaded region is a _____________ ___ _____ _________________.
… or IF the y is on the left,
“
than”
“less than
”
“greater than”
“
or equal to”
Example Problem
line
line
line
line
shade
shade
shade
shade
Steps
y  x 1
1st: Make sure the inequality is in
________ - _____________ form
y
2nd: Using the _____________ and ______, graph the line
x
3rd: Determine if the line is ____________ or __________
4th: Shade the _____________ region
37
Algebra 1
Mrs. Bondi
Unit 4 Notes: Function Characteristics
Practice:
Graph the linear inequality.
y  2 x  4
A.
B.
2 x  5 y  10
Solve the following by writing an inequality and graphing it.
Suppose you intend to spend no more than $12 on peanuts and cashews for a party. How many pounds of
each can you buy if peanuts are $2/lb and cashews are $4/lb?
Which of the following are possible solutions?
a)
b)
c)
d)
6 lb. peanuts, no cashews
1 lb. peanuts, 3 lb. cashews
2 lb. peanuts, 1 lb. cashews
1.5 lb. peanuts, 2 lb. cashews
Write an inequality to match the graph.
HW: p.393 #8-28 even, 32-34, 37, 42-43
(need 5 coordinate grids)
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Ch.5 Review
5.1, 3-7