college algebra & trigonometry i - freehqasmi.free.fr/math1100/section2.5.pdf · 2007-04-10 ·...
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OutlineGraphs of Common Functions
Transformation
College Algebra & Trigonometry I
2.5 - Transformations on Functions
Math 1100
April 8, 2007
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
1 Graphs of Common FunctionsQuadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
2 TransformationVertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Quadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
Common Graphs
0 1 2 3 4 5−1−2−3−4−5
0
1
2
3
4
5
−1
−2
−3
−4
−5
Figure: Standard Quadratic Function y = x2
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Quadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
Common Graphs
0 1 2 3 4 5−1−2−3−4−5
0
1
2
3
4
5
−1
−2
−3
−4
−5
Figure: Square Root Function y =√
x
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Quadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
Common Graphs
0 1 2 3 4 5−1−2−3−4−5
0
1
2
3
4
5
−1
−2
−3
−4
−5
Figure: Constant Function y = 3
2
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Quadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
Common Graphs
0 1 2 3 4 5−1−2−3−4−5
0
1
2
3
4
5
−1
−2
−3
−4
−5
Figure: Absolute Value Function y = |x |
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Quadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
Common Graphs
0 1 2 3 4 5−1−2−3−4−5
0
1
2
3
4
5
−1
−2
−3
−4
−5
Figure: Cubic Function y = x3
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Quadratic FunctionSquare Root FunctionConstant FunctionAbsolute Value FunctionCubic FunctionCubic Root Function
Common Graphs
0 1 2 3 4 5−1−2−3−4−50
1
2
3
4
5
−1
−2
−3
−4
−5
Figure: Cubic Root Function y = 3√
x
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Vertical Shifts
Theorem
Let f be a function and c > 0.
The graph of y = f (x) + c is
the graph of y = f (x) shifted c
units vertically upward.
The graph of y = f (x) − c is
the graph of y = f (x) shifted c
units vertically downward.O
c
y = f (x) + c
y = f (x)
x
y
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) = |x | to graph g(x) = |x | − 4
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Horizontal Shifts
Theorem
Let f be a function and c > 0.
The graph of y = f (x + c) is
the graph of y = f (x) shifted c
units horizontaly to the left.
The graph of y = f (x − c) is
the graph of y = f (x) shifted c
units horizontaly to the right.O
c
y = f (x)
y = f (x − c)
x
y
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) = x2 to obtain the graph of g(x) = (x + 3)2
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Combining Horizontal and Vertical Shifts
Example
Use the graph of f (x) = x3 to graph g(x) = (x − 1)3 − 1
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Reflections
Theorem
The graph of y = −f (x) is the graph of y = f (x) reflected about
the x-axis.
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) =√
x to obtain the graph of g(x) = −√x
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Reflections
Theorem
The graph of y = f (−x) is the graph of y = f (x) reflected about
the y-axis.
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) = x + 1 to obtain the graph ofg(x) = −x + 1
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Vertical Stretching and Shrinking
Theorem
Let f be a function and c > 0.
If c > 1, then the graph of y = cf (x) is the graph of
y = f (x) vertically stretched by multiplying each of its
y-coordinates by c.
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Vertical Stretching and Shrinking
Theorem
Let f be a function and c > 0.
If c > 1, then the graph of y = cf (x) is the graph of
y = f (x) vertically stretched by multiplying each of its
y-coordinates by c.
If 0 < c < 1, then the graph of y = cf (x) is the graph of
y = f (x) vertically shrunk by multiplying each of its
y-coordinates by c.
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) = x3 to obtain the graph of
h(x) =1
2x3
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Horizontal Stretching and Shrinking
Theorem
Let f be a function and c > 0.
If c > 1, then the graph of y = f (cx) is the graph of y = f (x)horizontally shrunk by dividing each of its x-coordinates by c.
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Horizontal Stretching and Shrinking
Theorem
Let f be a function and c > 0.
If c > 1, then the graph of y = f (cx) is the graph of y = f (x)horizontally shrunk by dividing each of its x-coordinates by c.
If 0 < c < 1, then the graph of y = f (cx) is the graph of
y = f (x) horizontally stretched by dividing each of its
x-coordinates by c.
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of y = x 2 to obtain the graph of
1 g(x) = f (2x)
2 h(x) = f (1
2x)
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Sequence of Transformations
One can considers a function encountering more than onetransformation in the following order:
1 Horizontal Shifting
2 Stretching or Shrinking
3 Reflecting
4 Vertical Shifting
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) = x2 to obtain the graph of
g(x) = 2(x + 3)2 − 1
Math 1100 College Algebra & Trigonometry I
OutlineGraphs of Common Functions
Transformation
Vertical ShiftsHorizontal ShiftsCombining Horizontal and Vertical ShiftsReflectionsStretching and ShrinkingSequence of Transformations
Example
Use the graph of f (x) = 3√
x to obtain the graph of
g(x) = −1
23√
x − 1 + 3
Math 1100 College Algebra & Trigonometry I