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Determining the Key Features of Function Graphs 10 February 2011

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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

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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

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

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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.

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Examples Domain:

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Example Domain:

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Determining Domain - Infinity Domain:

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Examples Domain:

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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.

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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

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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.

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Examples Range:

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Examples Range:

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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.

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

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

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

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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!

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

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Function Name: Linear Parent Function: f(x) = x “Baby” Functions: f(x) = 3x f(x) = x + 6 f(x) = –4x – 2 y x2 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) = x 2 f(x) = 5x 2 – 14 f(x) = f(x) = f(x) = x 3 f(x) = -2x 3 + 4x 2 – x + 2

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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 1. Special Operation (absolute value, greatest integer) 2. Division by x 3. Highest Exponent (this includes square roots and cube roots)

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Examples 1. f(x) = x 3 + 4x – 3 2. f(x) = -2| x | + 11 3.

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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 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!

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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) =

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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

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Example 1. Find f(1) 2. Find f(–0.5) 3. Find f(x) = 0 4. 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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