Lesson 1: 2.1 Symmetry (3-1)

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Unit 2 Graph-itti!

• Lesson 1 2.1 Symmetry (3-1)
• Lesson 2 2.2 Graph Families (3-2, 3-3)
• Lesson 3 2.3 Inverses (3-4)
• Lesson 4 2.4 Continuity (3-5)
• Lesson 5 2.5 Extrema (3-6)
• Lesson 6 2.6 Rational Functions (3-7)

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Warm-up
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Unit Two Graph-itti

• In this unit we will learn
• STANDARD 2.1 use algebraic tests to determine
  symmetry in graphs, including even-odd tests
  (3-1)
• STANDARD 2.2 graph parent functions and
  perform transformations to them (3-2, 3-3)
• STANDARD 2.3 determine and graph inverses of
  functions (3-4)
• STANDARD 2.4 determine the continuity and end
  behavior of functions (3-5)
• STANDARD 2.5 use appropriate mathematical
  terminology to describe the behavior of graphs
  (3-6)
• STANDARD 2.6 graph rational functions (3-7)

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STANDARD 2.1 use algebraic tests to determine
symmetry in graphs, including even-odd tests
(3-1)

• In this lesson we will
• Discuss what symmetry is and the different types
  that exist.
• Learn to determine symmetry in graphs.
• Classify functions as even or odd.

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What is Symmetry?

• Point Symmetry Symmetry about one point
• Figure will spin about the point and land on
  itself in less than 360º.

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Formal Definition
P
M
P
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Symmetry to Origin

• This is the main point we look at for symmetry.
• Lets build some symmetry!

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Determining Symmetry with Respect to the Origin
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Lets do a couple
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Line Symmetry
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Whats that mean?
U D
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Lines We Are Interested In

• x-axis
• y-axis
• y x
• y -x

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x-axis
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y-axis
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y x
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y -x
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Checking Mathematically
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Homework

• HW 2.1 P 134 15 35 odd

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Warm-up

• Get a piece of graph paper and a calculator.
• Graph the following on separate axii

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Homework
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STANDARD 2.2 graph parent functions and perform
transformations to them (3-2)

• In this section we will
• Identify the graphs of some simple functions.
• Recognize and perform transformations of simple
  graphs.
• Sketch graphs of related functions.

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Families of Graphs

• Any function based on a simple function will have
  the basic look of that family.
• Multiplying, dividing, adding or subtracting from
  the function may move it, shrink it or stretch it
  but wont change its basic shape.

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Example
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Reflections
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Vertical Translations
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Horizontal Translations
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Vertical Dilations
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Horizontal Dilations
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Lets do some
Send One person from your group to get a white
board with a graph on it, a pen and an eraser.
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STANDARD 2.2 graph parent functions and perform
transformations to them (3-3)

• In this section we will
• Use function families to graph inequalities.

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So how do I graph this?
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Homework

• HW1 2.2 P 143 13-29 odd, 33
• HW2 2.2 P 150 21-31 odd

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Warm-up
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Homework
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STANDARD 2.3 determine and graph inverses of
functions (3-4)

• In this section we will
• Determine inverses of relations and functions.
• Graph functions and their inverses.

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Inverse Relations

• An inverse of function will take the answers
  (range) from the function and give back the
  original domain.

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Finding the inverse of a relation

• Easy!!! Just switch the domain and range!
• Are they both functions?

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Property of Inverse Functions

• If f(x) and f 1(x) are inverse functions, then
• In other words
• Two relations are inverse relations iff one
  relation contains the element (b,a) whenever the
  other relation contains (a,b).
• Does this remind you of something?

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Lets look at them on a graph
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A function and its inverse

• Are reflections of each other over the line y x.

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Is the inverse a function?
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Quick and dirty test

• If the original function passes the HORIZONTAL
  line test then the inverse will be a function.
• Lets check our parent graphs.

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Is the inverse a function?
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Is this a function?
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Proving Inverses

• If two functions are actually inverses then both
  the composites of the functions will equal x.
• You must prove BOTH true.

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Check to see if the following are inverse
functions
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The Handy, Dandy Build Your Own Inverse Kit

• Replace f(x) with y (it is just easier to look at
  this way).
• Switch the x and y in the equation.
• Resolve the equation for y.
• The result is the inverse.
• Now check!

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Try this
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Word Problem

• The fixed costs for manufacturing a particular
  stereo system are 96,000, and the variable costs
  are 80 per unit.
• A. Write an equation that expresses the total
  cost C(x) as a function of x given that x units
  are manufactured.

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• B. Determine the equation for the inverse process
  and describe the real-world situation it models.

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• C. Determine the number of units that can be
  made for 144,000.

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Homework

• HW 2.3 P 156 15 39 odd and 45

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Warm-up
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Homework
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STANDARD 2.4 determine the continuity and end
behavior of functions (3-5)

• In this section we will
• Determine the continuity or discontinuity of a
  function.
• Identify the end behavior of functions.
• Determine whether a function is increasing or
  decreasing on an interval.

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Continuity

• A continuous functions graph can be drawn
  without ever lifting up your pencil.
• It has no holes or gaps.

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Discontinuities

• Anything which disrupts the flow of the graph.
• What parent graphs do we have which demonstrate
  discontinuous functions?

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Types of discontinuities p 159

• Function is undefined at a value but, otherwise,
  the graph matches up.
• Graph has a hole.

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Types of discontinuities p 159

• Graph stops at one y-value, then jumps to a
  different y-value for the same x-value.
• Common in piece-wise functions.

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Types of discontinuities p 159

• A major disruption in the graph.
• As graph approaches the domain restriction, the
  graph will shoot towards either positive or
  negative infinity.

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Continuity at a single value
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Continuous? If not, what type of discontinuity
exists?
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Continuity on an interval

• A function is continuous on an interval iff it is
  continuous at each number in the interval.

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Increasing and decreasing functions
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AhhhhHuh?

• Increasing means uphill left to right.
• Decreasing means downhill left to right.
• Constant means a flat or horizontal line left to
  right.

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Behavior over an interval










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Graph these on your calculator

• P 166 26, 28, 30
• Determine the intervals where the functions are
  increasing or decreasing.
• Write the intervals in interval notation and in
  in terms of x.

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Answers
26.
28.
30.
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End Behavior

• What will the function be doing at the outermost
  reaches of its domain and range?

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Check these out!
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Homework

• HW 2.4 P 166 13 31 odd, 39
• You will need a graphing calculator.

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Check these out! Find discontinuities, intervals
inc/dec and end behavior
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Homework
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STANDARD 2.5 use appropriate mathematical
terminology to describe the behavior of graphs
(3-6)

• In this section we will
• Find the extrema of functions.
• Learn the difference between Absolute Extrema and
  Relative Extrema.
• Find the point of inflection of a functions
  (if it exists).

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Extrema Minimums and Maximums

• An Absolute Minimum of Maximum is the lowest or
  highest value the range of the function can have.
• The slope of the line drawn tangent to the min or
  max will have a slope of zero.
• That point is called a critical point for the
  graph.

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Find the Extrema and Describe End Behavior
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Relative Minimums and Maximums

• These points are not the absolute highs or lows
  for the function but they are the high or low
  over a certain interval.
• The slope of the line tangent to a relative min
  or max is still zero so the point is a critical
  point.
• Minimums are said to be concave up and maximums
  are concave down.

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Find the extrema
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Points of Inflection

• A point of inflection occurs when a graph changes
  from one concavity to another.
• The slope of the tangent line to this point is
  undefined ( a vertical line). This point is also
  considered a critical point.
• You will learn to calculate the point of
  inflection in calculus.

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Determine whether there is a minimum, a maximum
or a point of inflection at the given point
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Homework

• HW 2.5 P 177 13 29 every other odd and 34
• You will need a graphing calculator.

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Describe the end behavior of the graph.
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STANDARD 2.6 graph rational functions (3-7)

• In this section we will
• Graph Rational Functions
• Determine vertical, horizontal and slant
  asymptotes

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Rational Functions

• Have a variable in the denominator.
• The denominator restriction will have a profound
  effect on the functions graph.

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Vertical Asymptotes and Holes

• Caused by values which make the denominator 0.
• Also known as removable and non-removable
  discontinuities.

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Holes or Removable Discontinuities
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Vertical Asymptotes
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Building a function
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Horizontal and Slant Asymptotes

• 3 cases possible
• Degree of numerator lt Degree of denominator H.A.
  at y 0.
• Degree of numerator Degree of denominator H.A.
  is the ratio of the coefficients.
• Degree of numerator gt Degree of denominator Do
  long division to find the Slant Asymptote.

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Examples
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Homework

• HW 2.6 P 186 15 39 odd, 43