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Chapter 1: Functions
Vogler
Algebra II
Functions
Functions give a one to one relationship
between two variables:
Y=2x, z=5+u, Pnuts+Bter=PB
 You get the idea.
 We can tell that a graph is a function by
using the vertical line test:

Functions
Functions

Since functions give a one to one
relationship between two variables, then
we can also identify functions on tables:
There it is!!
x
2
4
6
8
y
1
7
9
10
Functions

Domain: all possible values of x.


Range: all possible values of y.


Stated as: all reals or with notation
{∞<x<∞}
Stated as: all reals or with notation
{∞<x<∞}
The domain and range depend on the
graph.
Types of Functions
16
18
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
1
2
3
4
-5
5
0
Quadratic
3.5
Linear
3
2.5
2
1.5
1
0.5
Absolute
0
-4
-2
5
0
2
4
Functions

Function notation is just a fancy way to write
y=:




y=3x-7
f(x)=3x-7
t: x3x-7
It gives us a short hand way of telling us to
evaluate functions for certain numbers:



Evaluate y=x2+4 for x=3
f(3)=x2+4
t: 3x2+4
Distributive Property
The distributive property helps us to
simplify
 a(b+c)=ab+bc
 4(2+x)
4•2+4x
8+4x

Simplifying Expressions
Simplified
expressions are
easier:
4x+3x+2+1


7x+3
Combine like terms
Apply the
distributive property:
7(3x+4)+5
21x+28+5
21x+33

Solving Multi-Step Equations
2 cars are traveling towards
each other. Car A is going
50 MPH. Car B is going 60
MPH. They started 120
miles from each other, how
long before they pass?

The closing speed is
an increase:



Add 50+60
110 MPH
d=rt formula:
110t=120
 T=120/110
 1.09 hours (about 1
hour and 6 minutes)
Car B (60 MPH)

Car A (50 MPH)
Solving Multi-Step Equations
2 cars are traveling towards
each other. Car A is going
50 MPH. Car B is going 60
MPH. They started 120
miles from each other, how
long before they pass?
Algebraically:
50t+60t=120
 Combine like terms:
110t=120
 Divide:
t=120/110
t=1.09 hours

Solving Multi-Step Equations
Method 1:
2(3x+4)=20
Since 2 goes into 20 evenly,
divide it first:
3x+4=10

Method 2:
2(3x+4)=20
Distributive property
6x+8=20

3x=6
6x=12
x=2
x=2
Clearing Fractions
Sometimes, fractions suck…
 Get rid of them, multiply by the reciprocal:
X/4=20
Multiply both sides by 4
4(x/4)=4(20)
X=80

Solving Multi-Step Equations

Same direction travel is a
decrease:

A
B
Two runners are running in
the same direction. Runner
A is going 10 MPH. Runner
B is going 8 MPH and is 0.5
miles ahead. How long
does it take runner A to
overtake runner B?

Runner A is running 2 MPH
faster than B:


Think d=rt:



10-8=2
2t=0.5
t=0.5/2
Decimals? No problem:

t=5/20 or 1/4 of an hour
2 MPH
Solving Multi-Step Equations
Two runners are
running in the same
direction. Runner A
is going 10 MPH.
Runner B is going 8
MPH and is 0.5
miles ahead. How
long does it take
runner A to overtake
runner B?


Algebraically
10t=8t+0.5
Combine like terms:
10t-8t=8t-8t+0.5
2t=0.5
2t/2=0.5/2
t=5/20
t=1/4
Solving equations

Work backwards
through the order of
operations:
1.
2.
3.
4.



Parentheses
Exponents
Multiply/Divide
Add/Subtract
Combine like terms
Clear fractions
Apply the distributive
property

You made $250.
Each hour, you
make $2.90 of base
pay plus about
$32.50 in tips. You
also get a bonus of
$50 if you make
more than $200.
Solving equations cont.

Create an equation and solve





250=h(2.90+32.50)+50
200=h(35.4)
5.6=h
You worked 5.6 hours.
In word problems, use context to check.
Sequences: explicit and
recursive formulas




Recursive formulas refer to previous
numbers in a pattern to find the next
one: 2, 4, 6, 8, …
First term: a1
All other terms: an
Therefore:


a1=2
an=an-1+2
Sequences cont.

Explicit formulas give the resulting number
from a pattern without regard for previous
numbers: 2, 4, 6, 8, …
Term
Value



1
2
2
4
3
6
a(n)=2n
The difference between terms is the slope in
the equation y=mx+b
b=the term before the 1st term
Sequences

Write the recursive and explicit formulas
for the following sequence: 8, 12, 16…


Find the first term: a1
Find the difference between each
successive term