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.

Stated as: all reals or with notation {∞<x<∞}

Range: all possible values of y. Stated as: all reals or with notation

{∞<x<∞} The domain and range depend on the

graph.

Types of Functions

0

2

4

6

8

10

12

14

16

1 2 3 4 5

0

2

4

6

8

10

12

14

16

18

-5 0 5

0

0.5

1

1.5

2

2.5

3

3.5

-4 -2 0 2 4

Linear Quadratic

Absolute

Functions Function notation is just a fancy way to write

y=: y=3x-7 f(x)=3x-7 t: x3x-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: 3x2+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 Equations2 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 A (50 MPH)

Car B (60 MPH)

Solving Multi-Step Equations2 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 3x=6 x=2

Method 2:

2(3x+4)=20

Distributive property

6x+8=20

6x=12

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: 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:

10-8=2 Think d=rt:

2t=0.5 t=0.5/2

Decimals? No problem: t=5/20 or 1/4 of an hour

AB 2 MPH

Solving Multi-Step EquationsTwo 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. Parentheses2. Exponents3. Multiply/Divide4. 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, …

a(n)=2n The difference between terms is the slope in the

equation y=mx+b b=the term before the 1st term

Term 1 2 3

Value 2 4 6

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