Characteristics of Functions

Types of Functions Continuous has NO breaks Discrete has gaps or breaks

Domain & Range The domain of a relation is the set of all inputs or x-coordinates. The range of a relation is the set of all outputs or y-coordinates.

Notation Set Notation If the graph is discrete, list all of the inputs or outputs inside the squiggly brackets. Example: D= {1,2,4,5,7}

Notation Interval Notation For each continuous section of the graph, write the starting and ending point separated by a comma. Parenthesis: point is not included in Domain/ Range Brackets: point is included in Domain/ Range Start End

Notation Algebraic Notation Use equality and inequality symbols and variables to describe the domain and range. Examples: y > 5

Intercepts x-intercept – the point at which the line intersects the x-axis at (x, 0) y-intercept – the point at which the line intersects the y-axis at (0, y)

Find the x and y intercepts, then graph. -3x + 2y = 12

Find the x and y intercepts, then graph. 4x - 5y = 20

Increasing, Decreasing, or Constant Sweep from left to right and notice what happens to the y-values Finger Test- as you move your finger from left to right is it going up or down? Increasing goes up (L to R) Decreasing falls down (L to R) Constant is a horizontal graph

Characteristics Domain: Range: Intercepts: Increasing or Decreasing?

Characteristics Domain: Range: Intercepts: Increasing or Decreasing?

Just for practice, let’s look at an example that is NOT a linear function Domain: Range: Intercepts:

Average Rate of Change

Rate of Change A ratio that describes how one quantity changes as another quantity changes We know it as slope.

Rate of Change Positive – increases over time Negative – decreases over time Zero- doesn’t change over time Horizontal movement Undefined - vertical movement

Rate of Change Linear functions have a constant rate of change, meaning values increase or decrease at the SAME rate over a period of time.

Average Rate of Change using function notation

Ex 1 Find the Average Rate of Change f(x) = 2x2 – 3 from [2, 4].

Ex 2 Find the Average Rate of Change f(x) = -4x + 10 from [-1, 3]. m = -4

A. Find the rate of change from day 1 to 2. Ex 3 Find the Average Rate of Change A. Find the rate of change from day 1 to 2. m = 11 Days (x) Amount of Bacteria f(x) 1 19 2 30 3 48 4 76 5 121 6 192 B. Find the rate of change from day 2 to 5.

Ex 4 Find the Average Rate of Change Households in Millions f(x) In 2008, about 66 million U.S. households had both landline phones & cell phones. Find the rate of change from 2008 – 2011. Year (x) Households in Millions f(x) 2008 66 2009 61 2010 56 2011 51 m = -5 What does this mean? It decreased 5 million households per year from 2008 – 11.

Characteristics of Functions Classwork Characteristics of Functions

Characteristics of Functions Homework Characteristics of Functions