Functions

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Functions

• Our objectives
• Recognize Parent Functions
• Graphically Algebraically
• Please take notes and ALWAYS ask questions ?

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Pre-Calculus 2

• Todays Objectives

Todays Agenda

• Homework

Do NOW Define domain and range in your
notebook. 10 mins
.
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The following basic graphs will be used
extensively in this section. It is important to
be able to sketch these from memory.
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Constant Functionf(x) a
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Linear function f(x) x
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quadratic function
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cubic function
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Polynomial Function

• http//zonalandeducation.com/mmts/functionInstitut
  e/polynomialFunctions/graphs/polynomialFunctionGra
  phs.html
• zero degree
• first Degree
• second degree
• third degree
• Fourth degree

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Exponential Functionf(x) a
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Logarithmic Functionf(x)log x
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square root function
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cube root function
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absolute value function
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Rational Functionf(x)
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Reciprocal Functionf(x)
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Inverse Function
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Piece-wise Function
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Piece-wise Function

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We will now see how certain transformations
(operations) of a function change its graph. This
will give us a better idea of how to quickly
sketch the graph of certain functions. The
transformations are (1) translations, (2)
reflections, and (3) stretching.
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Vertical Translation

• Vertical Translation
• For b gt 0,
• the graph of y f(x) b is the graph of y
  f(x) shifted up b units
• the graph of y f(x) ? b is the graph of y
  f(x) shifted down b units.

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Horizontal Translation

• Horizontal Translation
• For d gt 0,
• the graph of y f(x ? d) is the graph of y
  f(x) shifted right d units
• the graph of y f(x d) is the graph of y
  f(x) shifted left d units.

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• Vertical shifts
• Moves the graph up or down
• Impacts only the y values of the function
• No changes are made to the x values
• Horizontal shifts
• Moves the graph left or right
• Impacts only the x values of the function
• No changes are made to the y values

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The values that translate the graph of a function
will occur as a number added or subtracted either
inside or outside a function.Numbers added or
subtracted inside translate left or right, while
numbers added or subtracted outside translate up
or down.
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Recognizing the shift from the equation, examples
of shifting the function f(x)

• Vertical shift of 3 units up
• Horizontal shift of 3 units left (HINT xs go
  the opposite direction that you might believe.)

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Points represented by (x , y) on the graph of
f(x) become
If the point (6, -3) is on the graph of
f(x), find the corresponding point on the graph
of f(x3) 2
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Combining a vertical horizontal shift

• Example of function that is shifted down 4 units
  and right 6 units from the original function.

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Reflections

• The graph of ?f(x) is the reflection of the
  graph of f(x) across the x-axis.
• The graph of f(?x) is the reflection of the
  graph of f(x) across the y-axis.
• If a point (x, y) is on the graph of f(x), then
• (x, ?y) is on the graph of ?f(x), and
• (?x, y) is on the graph of f(?x).

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Reflecting

• Across x-axis (y becomes negative, -f(x))
• Across y-axis (x becomes negative, f(-x))

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Vertical Stretching and Shrinking

• The graph of af(x) can be obtained from the
  graph of f(x) by
• stretching vertically for a gt 1, or
• shrinking vertically for 0 lt a lt 1.
• For a lt 0, the graph is also reflected across the
  x-axis.
• (The y-coordinates of the graph of y af(x) can
  be obtained by multiplying the y-coordinates of y
  f(x) by a.)

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VERTICAL STRETCH (SHRINK)

• ys do what we think they should If you see
  3(f(x)), all ys are MULTIPLIED by 3 (its now 3
  times as high or low!)

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Horizontal stretch shrink

• Were MULTIPLYING by an integer (not 1 or 0).
• xs do the opposite of what we think they should.
  (If you see 3x in the equation where it used to
  be an x, you DIVIDE all xs by 3, thus its
  compressed horizontally.)