Determining the Key Features of Function Graphs 10 February 2011.
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Transcript of Determining the Key Features of Function Graphs 10 February 2011.
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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 Domain1. Start at the origin2. Move along the x-axis until you find the
lowest possible x-value. This is your lower bound.
3. Return to the origin4. Move along the x-axis until you find your
highest possible x-value. This is your upper bound.
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Examples
Domain:Domain:
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Example
Domain:
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Determining Domain - Infinity
Domain:
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Examples
Domain: 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: Range:
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Examples
Range: 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: x-intercepts:
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Y-Intercepts Where the graph crosses the y-axis
y-intercepts: 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:-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!
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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 Functions1. Linear2. Absolute Value3. Greatest Integer4. Quadratic
5. Cubic6. Square Root7. Cube Root8. 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) = x2 f(x) = 5x2 – 14 f(x) = f(x) = f(x) = x3 f(x) = -2x3 + 4x2 – x + 2
x1
x24
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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 x3. Highest Exponent (this includes square roots
and cube roots)
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Examples1. f(x) = x3 + 4x – 3
2. f(x) = -2| x | + 11
3. ]]x[[)x(f 2
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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) PointsPeaks (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.