Determining the Key Features of Function

Graphs

The Key Features of Function Graphs - Preview Domain and Range x-intercepts and y-intercepts Intervals of increasing, decreasing, and

constant behavior Parent Equations Maxima and Minima

Domain Domain is the set of all possible input or

x-values To find the domain of the graph we look

at the x-axis of the graph

Determining Domain - Symbols Open Circle → Exclusive ( )

Closed Circle → Inclusive [ ]

Identifying the Domain1. Start at the far left of the graph.2. Move along the x-axis until you find the

lowest possible x-value of the graph. This is your lower bound. Note if you have a open circle or a closed circle.

3. Keep moving along the x-axis until you find your highest possible x-value of the graph. This is your upper bound. Note if you have a open circle or a closed circle.

Examples

Domain:Domain:

Example

Domain:

Determining Domain - Infinity

Domain:

Examples

Domain: Domain:

Your Turn: Complete problems 3, 7, and find the

domain of 9 and 10 on pg. 160 from the Xeroxed sheets

3. 7.

9. 10.

3. 7.

9. 10.

11. 12.

Range The set of all possible output or y-

values To 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

***Identifying the Range1. Start at the bottom of the graph.2. Move along the y-axis until you find the

lowest possible y-value of the graph. This is your lower bound. Note if you have a open circle or a closed circle.

3. Keep moving along the y-axis until you find your highest possible y-value of the graph. This is your upper bound. Note if you have a open circle or a closed circle.

Examples

Range: Range:

Examples

Range: Range:

Alternative Way to Identify the Range – This slide isn’t in your notes!

Your Turn: Complete problems 4, 8, and find the

range of 9 and 10 on pg. 160 from the Xeroxed sheets

4. 8.

9. 10.

4. 8.

9. 10.

11. 12.

Challenge – Not in your notes!Identify the domain and range

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

x-intercept Roots Zeros

Examples

x-intercepts: x-intercepts:

Y-Intercepts Where the graph crosses the y-axis

y-intercepts: y-intercepts:

Seek and Solve!!!

Types of Function Behavior 3 types:

Increasing Decreasing Constant

When determining the type of behavior, we always move from left to right on the graph

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:-1-3

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.

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 Functions1. Linear2. Absolute Value3. Greatest Integer4. Quadratic

5. Cubic6. Square Root7. Cube Root8. Reciprocal

Parent Function Flipbook

Function Name: Linear Parent Function: f(x) = x

“Baby” Functions:

y

x2

2

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

x1

x24

“Baby” Functions, cont. f(x) = |x|

x)x(f

Your Turn: Create your own “baby” functions in your

parent functions book.

Identifying Parent Functions From Equations:

Identify the most important operation1. Special Operation (absolute value, greatest

integer)2. Division by x3. Highest Exponent (this includes square

roots and cube roots)

Examples1. f(x) = x3 + 4x – 3

2. f(x) = -2| x | + 11

3. ]]x[[)x(f 2

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) PointsPeaks (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.

1. 2.

3. 4.

5. 6.

Reminder: Find f(#) and Find f(x) = x

Find f(#) 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(#)

1. Draw a (vertical) line at x = #

2. The intersection points are points where the graph = f(#)

f(1) = f(–2) =

Evaluating Graphs of Functions – Find f(x) = #

1. Draw a (horizontal) line at y = #

2. The intersection points are points where the graph is f(x) = #

f(x) = –2 f(x) = 2

Example

1. Find f(1)

2. Find f(–0.5)

3. Find f(x) = 0

4. Find f(x) = –5

Your Turn: Complete Parts A – D for problems 7 – 14

on The Key Features of Function Graphs – Part III handout.

7. 8.

9. 10.

11. 12.

13. 14.