Determining the Key Features of Function Graphs 10 February 2011

The Key Features of Function Graphs - Preview Domain Range x-intercepts y-intercept End Behavior Intervals of increasing, decreasing, and constant behavior Parent Equation Maxima and Minima

Domain Reminder: Domain is the set of all possible input or x-values When we find the domain of the graph we look at the x-axis of the graph

Determining Domain - Symbols Open Circle → Exclusive ( ) Closed Circle → Inclusive [ ]

Determining Domain Start at the origin Move along the x-axis until you find the lowest possible x-value. This is your lower bound. Return to the origin Move along the x-axis until you find your highest possible x-value. This is your upper bound.

Examples Domain: Domain:

Example Domain:

Determining Domain - Infinity

Examples Domain: Domain:

Your Turn: In the purple Precalculus textbooks, complete problems 3, 7, and find the domain of 9 and 10 on pg. 160 3. 7. 9. 10.

Range The set of all possible output or y-values When we find the range of the graph we look at the y-axis of the graph We also use open and closed circles for the range

Determining Range Start at the origin Move along the y-axis until you find the lowest possible y-value. This is your lower bound. Return to the origin Move along the y-axis until you find your highest possible y-value. This is your upper bound.

Examples Range: Range:

Examples Range: Range:

Your Turn: In the purple Precalculus textbooks, complete problems 4, 8, and find the domain of 9 and 10 on pg. 160 4. 8. 9. 10.

X-Intercepts Where the graph crosses the x-axis Has many names: Roots Zeros

Examples x-intercepts: x-intercepts:

Y-Intercepts Where the graph crosses the y-axis y-intercepts:

Seek and Solve!!!

Roller Coasters!!! Fujiyama in Japan

Types of Behavior – Increasing As x increases, y also increases Direct Relationship

Types of Behavior – Constant As x increases, y stays the same

Types of Behavior – Decreasing As x increases, y decreases Inverse Relationship

Identifying Intervals of Behavior We use interval notation The interval measures x-values. The type of behavior describes y-values. Increasing: [0, 4) The y-values are increasing the x-values are between 0 inclusive and 4 exclusive when

Identifying Intervals of Behavior Increasing: Constant: Decreasing: x 1 1

Identifying Intervals of Behavior, cont. Increasing: Constant: Decreasing: x -3 -1 Don’t get distracted by the arrows! Even though both of the arrows point “up”, the graph isn’t increasing at both ends of the graph!

Your Turn: Complete problems 1 – 4 on The Key Features of Function Graphs – Part II handout.

1. 2. 3. 4.

What do you think of when you hear the word parent?

Parent Function The most basic form of a type of function Determines the general shape of the graph

Basic Types of Parent Functions Linear Absolute Value Greatest Integer Quadratic Cubic Square Root Cube Root Reciprocal

Function Name: Linear Parent Function: f(x) = x “Baby” Functions: 2 2 x

Greatest Integer Function f(x) = [[x]] f(x) = int(x) Rounding function Always round down

“Baby” Functions Look and behave similarly to their parent functions To get a “baby” functions, add, subtract, multiply, and/or divide parent equations by (generally) constants f(x) = x2 f(x) = 5x2 – 14 f(x) = f(x) = f(x) = x3 f(x) = -2x3 + 4x2 – x + 2

Your Turn: Create your own “baby” functions in your parent functions book.

Identifying Parent Functions From Equations Identify the most important operation Special Operation (absolute value, greatest integer) Division by x Highest Exponent (this includes square roots and cube roots)

Examples f(x) = x3 + 4x – 3 f(x) = -2| x | + 11

Identifying Parent Equations From Graphs Try to match graphs to the closest parent function graph

Examples

Your Turn: Complete problems 5 – 12 on The Key Features of Function Graphs handout

Maximum (Maxima) and Minimum (Minima) Points Peaks (or hills) are your maximum points Valleys are your minimum points

Identifying Minimum and Maximum Points Write the answers as points You can have any combination of min and max points Minimum: Maximum:

Examples

Your Turn: Complete problems 1 – 6 on The Key Features of Function Graphs – Part III handout.

Reminder: Find f(#) and Find f(x) = x Find the value of f(x) when x equals #. Solve for f(x) or y! Find f(x) = # Find the value of x when f(x) equals #. Solve for x!

Evaluating Graphs of Functions – Find f(#) Draw a (vertical) line at x = # The intersection points are points where the graph = f(#) f(1) = f(–2) =

Evaluating Graphs of Functions – Find f(x) = # Draw a (horizontal) line at y = # The intersection points are points where the graph is f(x) = # f(–2) = f(2) =

Example Find f(1) Find f(–0.5) Find f(0) Find f(–5)

Your Turn: Complete Parts A – D for problems 7 – 14 on The Key Features of Function Graphs – Part III handout.