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Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 1 of 18
Section 1 – Reading Graphs; Linear Equations in two variables
Notes: We sometimes abbreviate the condition x = 2 and y = 5 as (x, y) = (2, 5). Normally the
variables are listed in alphabetical order. As we wrote it, it is clear which variable is which. If not
explicitly indicated, the “ordered pair” is assumed to be in alphabetical order.
Homework: 9-14, 21-23, 48-50, 55-59, 69-71
Additional Questions
1) Which quadrant is each of the following ordered pairs in?
a) (1, 1)
b) (1, -1)
c) (-1, 1)
d) (-1, -1)
2) For each of the following equations, is the ordered pair (2, 3) a solution?
a) x + y = 5
b) 3x = 2y
c) 2x = 3y
3) Plot the point below on the graph to the right.
a) (4, 3)
b) (3, -4)
c) (-2, -2)
d) (-3, 4)
e) (2, 0)
f) (0, 2)
Complete each table of values, plot the ordered pairs and connect with a straight line.
4) 2x + 3y = 5
x
y
1
1
4
3
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 2 of 18
Section 2 – Graphing Linear Equations in two variables
Note: The book notes than any equation of the form Ax + By = 0 goes through the origin (0, 0).
You need another point to determine the line the equation satisfies. The points (B, -A) and (-B, A)
also lie on the solution line.
Homework: 4-6, 9-11, 19-21, 36
Additional Questions
1) For the equation y = 2x – 4 complete the ordered pairs below
and plot them on the graph to the right and draw the line through
them.
a) (0, ___)
b) (___, 0)
2) For the equation 3x + 2y = 12
a) What is the y-intercept?
b) What is the x-intercept?
3) Graph the equation 2y = 3x on the right.
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 3 of 18
Section 3 – Slope of a Line
Homework: 2-5, 13, 24-26, 33, 34, 36, 38, 46-50
Notes: We can use the two pictures below to help see slope. On the picture to the left is an
𝑎
𝑐
example of a positive slope of 𝑏 and the picture on the right has negative slope of − 𝑑. Later, you
𝑎
will learn that for the picture on the left, the lower left hand angle has tangent of 𝑏 so slope is
just a measure of the angle the line forms (with the x axis). Similarly for the triangle on the right,
𝑐
the tangent of the lower right angle is 𝑑. However, the tangent of the angle the line makes with
𝑐
the x-axis is − 𝑑.
Notes: For perpendicular lines, if one line is defined by the equation ax + by = c, all of its
perpendicular lines look like bx – ay = d. where a, b, c, and d, are constants. The numerical
coefficients are interchanged and one of the two numerical coefficients signs switches from
positive to negative or vice versa. Since multiplying an equation by a constant does not really
change the equation the perpendicular line may look like bnx – any = d where n is any non zero
constant. For parallel lines to ax + by = c, all of its parallel lines look like anx + bny = d.
Additional Questions
Find the slope of the straight lines going through the two points
1) (4, 1) and (5, 2)
2) (5, 2) and (4, 1)
3) (1, -5) and (-4, 3)
4) (-3, -2) and (2, 5)
Find the slope of the line defined by the equation
5) 4x + 5y = 7
6) 6x - 5y = 2
7) y = 4x + 5
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 4 of 18
Section 4 – Equations of Lines
Note: Below is how I prefer to see the standard forms of lines:
A. Point-slope: Given that a straight line passes through the point (𝑥1 , 𝑦1 ) with the slope m, the
equation for that line is: y - y1 = m(x - x1).
B. Two points: Given that a straight line passes through the two points (𝑥1 , 𝑦1 ) and (𝑥2, 𝑦2 ) the
equation for that line is:
C. Slope-intercept: Given that a straight line has slope m, and when x = 0, y = a, the equation for
that line is: y = mx + a.
D. Parallel line - point: Given that a straight line passes through the point (𝑥1 , 𝑦1 ) and is parallel
to the line ax + by = c, the new equation is 𝑎𝑥 + 𝑏𝑦 = 𝑎𝑥1 + 𝑏𝑦1.
E. Perpendicular line - point: Given that a straight line passes through the point (𝑥1 , 𝑦1 ) and is
perpendicular to the line ax + by = c, the new equation is 𝑏𝑥 − 𝑎𝑦 = 𝑏𝑥1 − 𝑎𝑦1 . NOTE: The
“-” means to flip the sign between the x and y term.
Homework: 3-6, 9-11, 13, 15-17, 22-24, 32-33, 37-40, 47-48
Additional Questions
2) Given: Slope = 2, y-intercept = (0,-3), write the equation of the line.
3) Change the equation 2x – 3y = 7 to slope intercept form
and graph the equation on the right.
4) Given: Slope = 2, point on line = (1,-1), write the equation of the line.
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 5 of 18
5) Given: Line passing through points (0, 4) and (-1, 3), find the equation of the line:
6) For the line 4x + 5y = 9:
a) Find the equation of the line parallel to the equation going through the point (3, -6)
b) Find the equation of the line perpendicular to the equation going through the point
(3, -6)
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 6 of 18
Section 5 – Graphing Linear Inequalities in two variables
Homework: 1, 5-7, 15-16, 19
Additional Questions
1) Graph 3x + 2y > 5.
2) Graph y > 2x – 2.
3) Graph 4x + 3y < 1.
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 7 of 18
4) Graph 3y – 2x < 4.
5) Graph x + 2y  7.
6) Graph x ≤ 3y.
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 8 of 18
Section 6 – Introduction to Functions
Homework: 1, 3, 5-7, 17-19, 23-25, 44-45, 51-52, 63-65
Additional Questions
For each of the following relations:
a) Is the relation a single valued function?
b) What is the range?
c) What is the domain?
1) 𝑦 = 𝑥 2 + 1
a)
b)
c)
2) 𝑥 = 𝑦 2 + 1
a)
b)
c)
3) 2𝑥 − 5𝑦 = 11
a)
b)
c)
For 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 2 − 3𝑥
4) Find 𝑓(2)
5) Find 𝑔(3)
6) Find 𝑓(𝑔(𝑥))
7) Find 𝑔(𝑓(𝑥))
Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 9 of 18
Vocabulary
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
axis
boundary
coincide
component
coordinate
dependent
diagram
domain
function
graphing
graphs
horizontal
independent
inequalities
intercept
interpret
notation
origin
parallel
perpendicular
plane
plotting
points
quadrants
range
rectangular
region
relation
rise
run
scatter
shade
slant
slope
solution
subscript
system
table (of values)
test
vertical
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 10 of 18
Section 1 – Reading Graphs; Linear Equations in two variables
Additional Answers
1) Which quadrant is each of the following ordered pairs in?
a) (1, 1)
I
b) (1, -1)
IV
c) (-1, 1)
II
d) (-1, -1)
III
2) For each of the following equations, is the ordered pair (2, 3) a solution?
a) x + y = 5
2+3=5
Yes
b) 3x = 2y
3(2) = 2(3)
6=6
Yes
c) 2x = 3y
Vocabulary
axis
coordinate
diagram
graphs
interpret
plane
points
quadrants
rectangular
scatter
solution
system
table (of values)
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 11 of 18
Section 2 – Graphing Linear Equations in two variables
Additional Answers
2) For the equation 3x + 2y = 12
c) What is the y-intercept?
d) What is the x-intercept?
Vocabulary
coincide
graphing
horizontal
intercept
origin
plotting
slope
vertical
(0, 6)
(4, 0)
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 12 of 18
Section 3 – Slope of a Line
Additional Answers
Find the slope of the straight lines going through the two points
1) (4, 1) and (5, 2)
2−1
1
Slope = 5−4 = 1 = 𝟏
2) (5, 2) and (4, 1)
1−2
−1
Slope = 4−5 = −1 = 𝟏
3) (1, -5) and (-4, 3)
3−(−5)
𝟖
𝟖
Slope = −4−1 = −𝟓 = − 𝟓
4) (-3, -2) and (2, 5)
5−(−2)
𝟕
Slope = 2−(−3) = 𝟓
𝟏
Find the slope of the line defined by the equation
5) 4x + 5y = 7
4
7
𝟒
𝑦 = − 5 𝑥 + 5 or slope = − 𝟓
6) 6x - 5y = 2
6
2
𝟔
𝑦 = 5 𝑥 − 5 or slope = 𝟓
7) y = 4x + 5 slope = 4
Vocabulary
rise
run
slant
slope
subscript
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 13 of 18
Section 4 – Equations of Lines
Additional Answers
2) Given: Slope = 2, y-intercept = (0,-3), write the equation of the line.
y = 2x – 3.
4) Given: Slope = 2, point on line = (1,-1), write the equation of the line.
y + 1 = 2(x – 1) or
y = 2x – 3
5) Given: Line passing through points (0, 4) and (-1, 3), find the equation of the line:
(y – 4)(-1 – 0) = (3 – 4)(x – 0)
-y + 4 = -x or
y=x+4
6) For the line 4x + 5y = 9:
a) Find the equation of the line parallel to the equation going through the point (3, -6)
4x + 5y = 4(3) + 5(-6) = 12 – 30 or 4x + 5y = – 18
b) Find the equation of the line perpendicular to the equation going through the point
(3, -6)
5x – 4y = 5(3) – 4(-6) = 15 + 24
or 5x – 4y = 39
Vocabulary
parallel
perpendicular
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 14 of 18
Section 5 – Graphing Linear Inequalities in two variables
Additional Answers
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 15 of 18
Vocabulary
boundary
inequalities
region
shade
test
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 16 of 18
Section 6 – Introduction to Functions
Additional Answers
For each of the following relations:
a) Is the relation a single valued function?
b) What is the range?
c) What is the domain?
1) 𝑦 = 𝑥 2 + 1
a) Yes
b) [1, ∞)
c) (-∞, ∞)
2) 𝑥 = 𝑦 2 + 1
a) No
b) (-∞, ∞)
c) [1, ∞)
3) 2𝑥 − 5𝑦 = 11
a) Yes
b) (-∞, ∞)
c) (-∞, ∞)
For 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 2 − 3𝑥
4) Find 𝑓(2) = 3(2) − 2 = 6 − 2 = 𝟒
5) Find 𝑔(3) = 2 − 3(3) = 2 − 9 = −𝟕
6) Find 𝑓(𝑔(𝑥)) = 3(2 − 3𝑥) − 2 = 6 − 9𝑥 − 2 = 𝟒 − 𝟗𝒙
7) Find 𝑔(𝑓(𝑥)) = 2 − 3(3𝑥 − 2) = 2 − 9𝑥 + 6 = 𝟖 − 𝟗𝒙
Vocabulary
component
dependant
domain
function
independent
notation
range
relation
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 17 of 18
Glossary
Axis: is a straight line where you measure one of the values from (on).
x-axis: is normally measures the x values from left (negative) to right (positive)
y-axis: is normally measures the y values from back (negative) to front (positive)
z-axis: is normally measures the z values from bottom (negative) to top (positive)
Boundary: of a figure is normally what separates the inside of a figure from the outside of the
figure. For an inequality, you might also consider it the line that separates where the inequality is
true from where the inequality is false.
Coincide: means at the same place. Examples: Two straight lines coincide (or meet or intersect)
at a point. Two straight lines that overlap each other are really the same line.
Component: Technically means part of the whole. In an ordered pair (or triplet, etc.) it is one of
the numbers that make up the ordered pair (or triplet, etc.).
Coordinate: When plotting a point it is one of the values used to place the point. Warning: For a
two dimensional graph, the x-coordinate is sometimes called the ordinate and the y-ordinate is
referred to as the coordinate.
Dependent (variable): A function of variables is called a dependent variable.
(Scatter) Diagram: Shows points on a Cartesian graph where data is been observed to occur.
Domain: is where a function is defined. May also be defined as the set of values where a
function is defined.
Horizontal: lines extend to your left and right.
Independent (variables): are used to define a function.
Inequalities: are used to limit something is true.
Intercept: is where a graph crosses a given axis (see axis above).
Notation: is a way of representing something.
Origin: of a Cartesian coordinate system is where all the coordinates are zero.
Parallel: straight lines are two lines that are at a constant distance from each other. (Pick a point
on one line and find the point closest to it on the other line. The distance between those two
points is always the same).
Perpendicular: lines meet at right angles to each other.
Plane: is a flat two-dimensional surface.
Plotting a point is determining where it goes on a graph.
Point: is a single location on a graph.
Quadrants: In the Cartesian plane, the x-axis and y-axis divide the plane into four quadrants.
Quadrant I: is the set of points where both the x-coordinate and y-coordinate are positive
Quadrant II: is the set of points where the x-coordinate is negative and the y-coordinate
is positive
Quadrant III: is the set of points where both the x-coordinate and y-coordinate are
negative
Quadrant IV: is the set of points where the x-coordinate is positive and the y-coordinate
is negative
Range: of a function is the set of values the function actually take on.
Rectangular coordinate system: Gives the location of points in the system by telling how to get
to that point by measuring several perpendicular distances from the origin.
Relation: Describes the constraints put upon several variables.
Rise: (see slope)
Run: (see slope)
Scatter: (See diagram)
Shade: is to color a figure to show where something is true.
Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 18 of 18
Slant: (see slope)
Slope: For a two dimensional graph is a measure of the slant of a line (or curve) at a particular
point on the curve. Basically it is the tangent of the angle that a straight line with the same slant
makes with the x-axis. This is sometimes referred to the rise over run (how much increase or
decrease the y-coordinate changes per change in the x-coordinate).
Solution: is one place where a condition or equation is true.
Table of Values: lists points on a graph. The table is used to help draw what the graph looks
like.
Test: Seeing if a particular condition is true or not.
Vertical: lines extend up and down.