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UNIT 1
Expressions, Equations and Functions
1-2: ORDER OF OPERATIONS
Students will . . .
evaluate numerical expressions by using the order of
operations.
evaluate algebraic expressions by using the order of
operations.
Order of Operations:
Parentheses
Exponents
Multiplication/Division (going left to right)
Addition/Subtraction (going left to right)
Please Excuse My Dear Aunt Sally
Example: Evaluate each of the following:
20 – 7 + 82 – 7 ∙ 11
48 ÷ 23 ∙ 3 + 5
4[12 ÷ (6 – 2)]2
25 6 2
33 5 3 2
Example: Evaluate the following algebraic
expressions for the given values:
a2(3b + 5) ÷ c if a = 2, b = 6, c = 4
2(x2 – y) + z2 if x = 4, y = 3, z = 2
Example: Ann is planning a business trip for
which she needs to rent a car. The car rental
company charges $38 per day plus $0.50 per mile
over 100 miles.
Write an expression for how much it will cost Ann to
rent the car.
Suppose Ann rents the car for 5 days and drives 180
miles. How much will she pay the rental company?
HW: Pg. 13 #16 – 38 even, 59
REVIEW
Evaluate each expression:
3∙5+1–2
14 ÷ 2 ∙ 6 – 52
3 9 2 32 9
39
1-4: THE DISTRIBUTIVE PROPERTY
Students will . . .
use the distributive property to evaluate expressions.
use the distributive property to simplify expressions.
Like terms are terms that have EXACTLY the
same variables.
Example: Identify the like terms below.
-4x3
2
2y2
-4x2
5x
-3
12x
-½
5x2
The coefficient of a term is the number in front of
the variable.
An expression is in simplest form when there are
no parentheses and the like terms are combined.
Example: Simplify each of the following:
6n – 4n
b2 + 13b + 13
4y3 + 2y – 8y + 5
7a + 4 – 6a2 – 2a
Distributive Property:
a(b + c) = ab + ac
(b + c)a = ba + ca
a(b - c) = ab - ac
(b - c)a = ba - ca
Example: Simplify each of the following:
2(4x + 8)
-6(r + 3g – t)
(y + 3)12
-4(-8 – 3m)
4(y2 + 8y + 2)
Example: Write and simplify an algebraic
expression for each of the following verbal
expressions:
5 times the difference of q squared and r plus 8 times
the sum of 3q and 2r
6 times the sum of x and y increased by four times
the difference of 5x and y
HW: Pg. 29 – 30 #12, 26 – 38 even, 56
REVIEW
Evaluate each expression:
23 – 2(17 + 33)
(12 3)
25 (16 3 5)
5
Simplify each expression:
(⅓ – 2b)27
4(8p + 4q – 7r)
4y3 + 3y3 + y4
4(x + 3) – 2x + 8
1-6: RELATIONS
Students will . . .
represent relations
interpret graphs of relations
A set or ordered pairs is called a relation.
The first number in the ordered pairs (x value) is
called the domain.
{(-1, 2) (-2, 3) (4, 5) (6, 7) (0, 1)}
D = {-2, -1, 0, 4, 6}
The second number in the ordered pairs (y value)
is called the range.
R = {1, 2, 3, 5, 7}
Example: For each of the following relations
identify the domain and range.
{ (0, 1) (2, 1) (3, 2) (3, 4) }
{ (2, 1) }
{ (4, 10) (1, 2) (3, 2) (2, 9) (5, 10) }
{ (1, 1) (-1, 1) }
Relation Representations
Ordered Pairs
{ (1, 1) (2, 3) (1, 3) (3, 4)}
Table
x coordinates in the first column and the corresponding y
coordinates in the second column (keep the pairs together)
x
y
1
1
1
3
2
3
3
4
Graph
Graph each point
Mapping
Make a list of x-values (in order with no repeats) and a list
of y-values (in order with no repeats); draw arrows to
corresponding values
Domain
Range
1
1
2
3
3
4
Example: Express each of the following relations
as a table, graph and mapping. Then determine
the domain and range.
{(4, -3), (3, 2), (-4, 1), (0, -3)}
{(4, 3), (-2, -1), (2, -4), (0, -4)}
In a relation, the independent variable is the
value that is subject to choice.
The dependent variable is the value that depends
on the independent variable.
represented by the x values
represented by the y values
Example: Identify the independent and
dependent variables for each relation.
The number of calories you burn increases as the
number of minutes that you walk increases
In warm climates, the average amount of electricity
used rises as the daily average temperature increases
and falls as the daily average temperature decreases.
Some relations can be interpreted by analyzing
their shape.
Example: For each of the following, describe what
happens in the graph.
HW: Pg. 43 – 45 #10 – 14 even, 15, 16, 19, 20, 40
REVIEW
Evaluate each expression:
7 + 2(9 – 3)
[(25 – 5) ÷ 9]11
Simplify each expression:
3(x – 6)
-5(2x + 9)
Express the relation as a table, graph and
mapping. Determine the domain and range.
{(-2, 4), (-1, 3), (0, 2), (-1, 2)}
1-7: FUNCTIONS
Students will . . .
determine whether a relation is a function
find function values
A function is a relationship between input and
output where there is exactly one output for each
input.
The x-values DO NOT repeat.
Example: Determine whether each relation is a
function.
Domain
Range
-2
1
0
7
4
{(2, 1), (3, -2), (3, 1), (2, -2)}
You can use the vertical line test to tell if a graph
is a function or not.
If a vertical line intersects the graph more than once,
then it is NOT a function.
Otherwise, it IS a function.
Example: Determine whether each graph
represents a function.
Equations that are functions can be written in
function notation.
Replace y with f(x)
“f of x”
Example: Write each of the following in function
notation.
y = 3x – 8
y = 3x2 – 4x + 10
A function with a graph that is not a straight line
is a nonlinear function.
If an equation has exponents that are not 1, then it is
a nonlinear function.
Example: If f(x) = 3x + 1 and g(x) = 2x2 – 5x – 6,
find each value:
f(-1)
g(0)
g(3) + 2
3[f(2)] + 7
g(-1) – 4[g(3)]
HW: Pg. 52 – 53 #20-25, 27, 28, 33-38
REVIEW
Evaluate:
3 + (5 – 2) – 32
35 ÷ 5 ∙ 22 + 1
Simplify each of the following:
3(2d + 6)
(6h – 1)5
Express the following relation as a table, graph,
and mapping. Determine the domain and range.
{(3, 1), (-1, 4), (-1, 7)}
REVIEW
Determine whether each relation is a function:
Domain
Range
-3
-1
1
3
7
-10
12
{(0,2), (3,5), (0,-1), (-2,4)}
42
If f(x) = 4x + 5, find each of the following:
f(3)
f(-5) + 1
1-8: INTERPRETING GRAPHS OF FUNCTIONS
Students will . . .
interpret intercepts, and symmetry of graphs and
functions.
interpret positive, negative, increasing, and
decreasing behavior, extrema, and end behavior of
graphs of functions.
The intercepts of a graph are the points where
the graph crosses an axis.
The point where the graph crosses the y-axis is the yintercept.
The point where the graph crosses the x-axis is the xintercept.
Example: The graph shows the
height y of an object as a
function of time x.
Is the function linear or
nonlinear?
Estimate the intercepts of the
graph.
What do these intercepts mean?
Example: The graph shows the temperature of a
medical sample thawed at a controlled rate.
Identify the function as linear or nonlinear.
Estimate and interpret the intercepts.
A graph has line symmetry if it can be divided
along a line into two parts that match exactly.
The line can be the y-axis or any other vertical line.
Example: An object is launched.
The graph shows the height y of
the object as a function of time x.
Does this graph have symmetry?
What is the line of symmetry?
What does this symmetry mean?
A function is positive where its graph is above
the x-axis.
A function is negative where its graph is below
the x-axis.
positive
negative
A function is increasing where the graph goes
up.
A function is decreasing where the graph goes
down.
High or low points of the graph are called
extrema.
Point A is a relative minimum since it has the lowest
y-coordinate of the points near it.
Point B is a relative maximum since it has the
highest y-coordinate of the points near it.
These can be found where the graph changes back
and forth between increasing and decreasing.
End behavior describes the values of a function
at the positive and negative extremes in its
domain.
As x decreases,
y increases.
As x increases,
y decreases.
Example: The graph shows the value of a vehicle
over time. Identify the function as linear or
nonlinear. Estimate and interpret the intercepts,
any symmetry, where the function is positive,
negative, increasing, and decreasing, the xcoordinates of any relative extrema, and the end
behavior of the graph.
HW: Pg. 59 #4 – 8 even, 11, 12, 18, 20
