1'2Elementary Functions Graphs and Transformations

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1.2 Elementary Functions Graphs and
Transformations

• In this presentation, you will be given an
  equation of a function and asked to draw its
  graph. You should be able to state how the graph
  is related to a standard function. It is not
  important that you plot a great many points for
  each graph. It IS important that you recognize
  the general shape of the graph. You can verify
  your answers using a graphing calculator, but
  only after you have attempted to construct the
  graph by hand.

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Problem 1

• Construct the graph of

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Solution
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Problem 2

• Now, sketch the related graph given by the
  equation below and explain, in words, how it is
  related to the first function you graphed.

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Solution Problem 2

• The graph has the same shape as the original
  function. The difference is that the original
  graph has been translated two units to the right
  on the x-axis. Conclusion The graph of the
  function f(x-2) is the graph of f(x) shifted
  horizontally two units to the right on the
  x-axis.
• Notice that replacing x by x-2 shifts the graph
  horizontally to the right and not the left.

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Problem 3

• Now, graph the following standard function
  Complete the table

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Solution to problem 3
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Problem 4

• Now, graph the following related function

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Solution to problem 4
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Problem 4 solution

• The graph of
• is obtained from the graph of
• by translating the graph of the original
  function up one unit vertically on the positive
  y-axis.

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Problem 5

• Graph
• What is the domain of this function?

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Solution to problem 5

• The domain is all non-negative real numbers. Here
  is the graph

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Problem 6

• Graph
• Explain, in words, how it compares to problem 5.

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Problem 6 solution(Notice that the graph lies
entirely within the fourth quadrant)
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Graph of f(x)

• The graph of the function f(x) is a reflection
  of the graph of f(x) across the x-axis. That is,
  if the graphs of f(x) and f(x) are folded along
  the x-axis, the two graphs would coincide.

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Cube root function

• Sketch the graph of the cube root function.
  Complete the table of ordered pairs

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Variation of cube root function

• Sketch the following variation of the cube root
  function

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Same graph as graph of cube root function.
Shifted horizontally to the left one unit.
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Graph of f(xc) compared to graph of f(x)

• The graph of f(xc) has the same shape as the
  graph of f(x) with the exception that the graph
  of f(xc) is translated horizontally to the left
  c units when c gt0 and is translated horizontally
  to the right c units when c lt 0.

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Absolute Value function

• Now, graph the absolute value function. Be sure
  to choose x values that are both positive and
  negative as well as zero.

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Graph of absolute value function

• Notice the symmetry of the graph.

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Variation of absolute value function
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Shift absolute value graph to the left one unit
and down two units on the vertical axis.