college algebra & trigonometry i - freehqasmi.free.fr/math1100/section2.5.pdf · 2007-04-10 ·...

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