Algebra IFunction Families

1. Linear – constant rate of change2. Absolute Value – distance from zero3. Rational 4. Square Root5. Quadratic6. Exponential

Geometric7. Power8. Trigonometric

Input:x-valueindependent variableindependent quantitydomaincause

Output:y-valuedependent variabledependent quantityrangeeffect

y = mx+ b

A linear relationshipm =slopeb =y-intercept, where x =0

Linear

Graph 4 Steps

1. TitleEquation or Words to Describe Graph

2. TableUse 5 values for x. Two negative, zero, and two positive

3. System Create a Cartesian Coordinate System with a scale to fit the data, and equal intervals

4. Line Should be neatly drawn to go through all points

y-intercept where x = 0x-intercept where y = 0

y-intercept

x-intercept

y-intercept where x = 0

x-intercept where y = 0

Slope

m =

Subscripts name the point Point 1 and Point 2(x1, y1) (x2 ,

y2)

Two points define a line.

2 1

2 1

y y

x x

y

x

Positive Slope A lot of work

Negative Slope

Wee!!!

Zero Slope No where fast…

Ski Bird on a horizontal hill.

UndefinedSlope

Divide by 0

Undefined!!!

Sheer doom awaits or does it?Ski Bird on a vertical hill.Where will this line end?

Slope Bear

Slope

Slope

UndefinedSlope

No change in x

Washington’s Mountaineering Club

This year, 4 climbers competed for the Steepmeister Cup. This award is given to the person who climbs the steepest slope. Each person went up Mount Kilimanjaro by a different route.

Determine who the winner is based on two points from each member's climbing log.

Climber 1 Desiree (30, 225), (55, 650)Climber 2 Mai (82, -35), (107, 565)Climber 3 Luis (125, 0), (125, 800)Climber 4 Kou (0, 10) (0, 1000)

Mount Kilimanjaro, East Africa•Compelling for its beauty•Highest peak on the African continent•Tallest free-standing mountain in the world•Found in isolation from the coastal area

Function• Matching Rule: for each x there is

only one y• Read f(x): “f of x” or “function of x”• Vertical Line Test – is a function if

touches graphed line only once

http://www.shodor.org/interactivate/activities/VerticalLineTest/

The steady acceleration of a car. (velocity vs. time)

Linear Function y=mx+b

V(t)

An airplane flies at an average speed. (distance vs. time)

Linear Function y=mx+b

The regular progress of the tortoise. (distance vs. time)

Linear Function y=mx+b

d(t)

Monthly deposits of equal paychecks. (amount vs. time)

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec

A bathtub fills at a constant rate. (volume vs. time)

Linear Function y=mx+b

V(t)

Problem

• Mrs. Kapler’s daughter has $100 in the bank.• She just got a job at the grocery store. It pays $8 per

hour. This daughter never spends her money; she prefers to save it all in the bank.

• Write an equation that represents the money she has in the bank as a function of hours worked.

• What is the slope? The y-intercept? • Create a table.• Graph the relationship. • After how many hours will she have $260 in the bank?

Sara has 4 hours after dinner to study and do her homework.

She has homework in math and English.

She spends x hours on math and y hours on English. Write an equation that describes this relationship.

Graph the equation.

If she spends 1 hour doing math, how much time does she have for English?

Vertical Line Test

FunctionsMatching RuleFor each x there is only one yEach element of the domain is paired with exactly one

element of the range. For each input (x) there is one and only one output (y)

f(x)x y

Function Notation

Output

InputName of Function

y f x

Is the relation a function?

1. {(2, 3), (3, 0), (5, 2), (4, 3)}

YES.

f(x)2 3

f(x)3 0

f(x)5 2

f(x)4 3

For each x there is only one y.

Is the relation a function?2. {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)}

f(x)4 1

f(x)5 2

f(x)5 3

f(x)6 6

f(x)1 9

No for 5 is paired with 2 and 3

For each x there is only one y.

Is this relation a function?{(1,3), (2,3), (3,3)}

1. Yes2. No

Vertical Line Test

If touches the graphed line only once is held true.

If held true, graphed line is a function. Are these functions?

FUNCTION! FUNCTION! NOPE!

Vertical Line Test

No Yes

Yes

No

Is this a graph of a function?

1. Yes2. No

Given f(x) = 3x - 2, find:1) f(3)

2) f(-2)

3(3)-23 7

3(-2)-2-2 -8

= 7

= -8

Given h(z) = z2 - 4z + 9, find h(-3)

(-3)2-4(-3)+9-3 30

9 + 12 + 9

h(-3) = 30

f(3) = 2(3) + 1 = 6 +1

= 71. 72. 13. -14. -7

Given f(x) = 2x + 1, find f(3)

Given g(x) = x2 – 2, find g(4)

1. 22. 63. 144. 18