Ordered pair: a set of independent (x) values that represent a relationship

Relation: a set of inputs (x) and outputs (y)

Function: a relation with one output (y) for each input (x) and no x value may be repeated

# Identifying Functions

With points: no x values can be the same or repeat

With a graph: must pass the vertical line test

→ Vertical line test: draw a vertical line through the graph, cutting through it. If it intersects at more than 1 point then it fails the test and thus is not a function

# Function Notation

The way that a function is written → f(x) is essentially our new y

f(x) can also be a value, f(value), and you would then substitute that value in for x

# Domain and Range of Functions

• Domain: every x value your function is allowed to be

• Range: every y value your function is allowed to be

Use:

< less than

less than or equal to

to set these boundaries of what your x and y values can be respectively

The quadratic function becomes a U-shaped parabola when graphed

### Forms:

• Standard Form:

f(x) = ax² + bx + c where a ≠ 0

• c is y intercept

• Cannot graph from this form

• Factored Form:

f(x) = a(x-r)(x-s)

• x intercepts are (r,0) and (s,0)

• To find the y intercept, let x = 0

• Can graph from this form

• Vertex Form:

f(x) = a(x-h)² + k

• Vertex is (h,k)

• To find y intercept, let x = 0

• Can graph from this form

### Direction of Opening, a’s Sign:

• If a is negative, parabola opens downwards and has a maximum (max frowns and is negative)

• If a is positive, parabola opens upwards and has a minimum

The min or max value is also the y value of the vertex

### Axis of Symmetry (AOS):

The value of the x value at the vertex of the parabola (at which the function is at its min or max value)

Formula: (r + s)/2

# Profit and Revenue

Revenue: the total amount of money taken in

Cost: the amount of money paid for goods or expenses

Profit: the amount of money left after expenses → revenue - cost

## Revenue

Revenue = selling price x number of items

1. Let x represent the number of [value] increases or decreases

2. Chart:

Old

New

Selling Price

Value

Old + or - [value]x

Number of Items

Value

Old + or - [value]x

If it is an increase, +

If it is a decrease, -

1. Find x intercepts → (new selling price)(new number of items) → solve by making it equal to zero

2. Find AOS by adding both points and dividing by 2 → (r+s)/2

3. Find new selling price or number of items by substituting in your new AOS value into either “new” equation on the chart (depending on what you want to find)

## Profit

Profit = profit per item x number of items

1. Let x represent the number of [value] increases or decreases

2. Chart:

Old

New

Profit Per Item

Value

Old + or - [value]x

Number of Items

Value

Old + or - [value]x

3. Find x intercepts → (Profit per item)(number of items) and solve by making x = 0

1. Find AOS by adding both points and dividing by 2 → (r+s)/2

2. Substitute back into chart “new” equations for new Profit Per Item or new Number of Items

• Factoring: Make the equation = to zero and factor, take your factored brackets and make them = to zero, isolate x

• Quadratic formula: make the equation = to zero and plug into the following formula

# Zeroes of a Quadratic Function

The radicand in quadratic formula is the discriminant → b² - 4ac

The value of the discriminant is an indication of the number of solutions

• Positive Discriminant: 2 solutions, real and unequal roots

• Discriminant = Zero: 1 solution, real and equal roots

• Negative Discriminant: no solutions, imaginary roots

# Discriminant with K

1. Find K by making D=0 → becomes a new quadratic

2. Draw a chart to find D>0 and D<0 (remember: the value of something that has been square root’ed is both positive and negative)

K

Pick and Plug, less than found (negative)

Found (negative)

Pick and plug, in between positive and negative

Found (positive)

Pick and plug, more than found (positive)

D

Calculate

0

Calculate

0

Calculate

``                               <                                  <                        <                                   <                       <``