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**Determining the Key Features of Function Graphs**

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**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

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**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

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**Determining Domain - Symbols**

Open Circle → Exclusive ( ) Closed Circle → Inclusive [ ]

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**Identifying the Domain**

Start at the far left of the graph. 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. 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.

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Examples Domain: Domain:

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Example Domain:

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**Determining Domain - Infinity**

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Examples Domain: Domain:

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Your Turn: Complete problems 3, 7, and find the domain of 9 and 10 on pg. 160 from the Xeroxed sheets

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**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

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**Identifying the Range Start at the far left of the graph.**

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. 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.

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Examples Range: Range:

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Examples Range: Range:

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Your Turn: Complete problems 4, 8, and find the range of 9 and 10 on pg. 160 from the Xeroxed sheets

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

Roots Zeros

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Examples x-intercepts: x-intercepts:

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

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Seek and Solve!!!

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**Types of Function Behavior**

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

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Roller Coasters!!! Fujiyama in Japan

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**Types of Behavior – Increasing**

As x increases, y also increases Direct Relationship

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**Types of Behavior – Constant**

As x increases, y stays the same

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**Types of Behavior – Decreasing**

As x increases, y decreases Inverse Relationship

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**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

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**Identifying Intervals of Behavior**

Increasing: Constant: Decreasing: x 1 1

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**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!

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

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1. 2. 3. 4.

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

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**Parent Function The most basic form of a type of function**

Determines the general shape of the graph

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**Basic Types of Parent Functions**

Linear Absolute Value Greatest Integer Quadratic Cubic Square Root Cube Root Reciprocal

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**Parent Function Flipbook**

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

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**Greatest Integer Function**

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

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**“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

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“Baby” Functions, cont. f(x) = |x|

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

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**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)

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

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**Identifying Parent Equations**

From Graphs: Try to match graphs to the closest parent function graph

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Examples

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

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**Maximum (Maxima) and Minimum (Minima) Points**

Peaks (or hills) are your maximum points Valleys are your minimum points

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**Identifying Minimum and Maximum Points**

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

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Examples

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

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**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!

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**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) =

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**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(x) = –2 f(x) = 2

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

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

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