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

Milestone Review A wild horse runs at a rate of 8 miles an hour for 6 hours. Let y be the distance, in miles, the horse travels for a given amount of time, x, in hours. This situation can be modeled by a function. Which of these describes the domain of the function? A. 0 ≤ x ≤ 6 B. 0 ≤ y ≤ 6 C. 0 ≤ x ≤ 48 D. 0 ≤ y ≤ 48

Milestone Review Let h(x) be the number of hours it takes a new factory to produce x engines. The company’s accountant determines that the number of hours it takes depends on the time it takes to set up the machinery and the number of engines to be completed. The relationship is modeled with the function h(x) = 6.5 + 5.25x. What is the domain of the function h(x) A) Negative numbers B) Whole numbers C) All positive numbers D) None of the above

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: Maximum & minimum

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.

Milestone Review A company uses the function V(x) = 28,000 – 1,750x to represent the amount left to pay on a truck, where V(x) is the amount left to pay on the truck, in dollars, and x is the number of months after its purchase. Use the table of values shown on next slide.

Milestone Review What is the y-intercept of the graph of the function in terms of the amount left to pay on the truck? Does the graph of the function have an x-intercept, and if so what does that represent? Does the function increase or decrease? x (months) V(x) ($) 28,000 1 26,250 2 24,500 3 22,750 4 21,000 5 19,250

Characteristics of Functions Classwork Characteristics of Functions