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 1. Start at the origin 2. Move along the x-axis until you find the lowest possible x-value. This is your lower bound. 3. Return to the origin 4. Move along the x-axis until you find your highest possible x-value. This is your upper bound.

Examples Domain:

Example Domain:

Determining Domain - Infinity Domain:

Examples Domain:

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

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:

Examples Range:

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

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

Examples 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 when the x-values are between 0 inclusive and 4 exclusive

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

Identifying Intervals of Behavior, cont.  Increasing:  Constant:  Decreasing: -3 y x 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.

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 1. Linear 2. Absolute Value 3. Greatest Integer 4. Quadratic 5. Cubic 6. Square Root 7. Cube Root 8. Reciprocal

Function Name: Linear  Parent Function: f(x) = x  “Baby” Functions: f(x) = 3x f(x) = x + 6 f(x) = –4x – 2 y x2 2

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