Math 170 - Section 2.3 - TopCatMath€¦ · Web view · 2016-06-103) 4) 5) 6) Translations ......

Math 170 - Cooley Pre-Calculus OCC

Section 2.3 – Families of Functions, Transformations, and Symmetry

Parent Functions – Here is the list of parent functions, f(x), that we are going to be using:

Square Function (Parabola) Square Root Function

Absolute Value Function Cubic Function

Cube-Root Function Greatest Integer Function

1

x f(x)–8 –2–6 –1.8–4 –1.6–2 –1.30 02 1.34 1.66 1.88 2

xf(x)–88–66–44–220022446688

xf(x)–4–64–3–27–2–8–1–

1001128327464

xf(x)–416–39–24–1100112439416

xf(x)00112 1.43 1.7425 2.26

2.47 2.68 2.8

xf(x)–1.9–2–1–1–0.9–1–0.4–

1000.400.90111.91


Section 2.3 – Families of Functions, Transformations, and SymmetryHere is a summary of the transformations of y = f ( x ).

1. Vertical translation: For , The graph of is the parent graph, , shifted up units; The graph of is the parent graph, , shifted down units.

2. Horizontal translation: For , The graph of is the parent graph, , shifted left units. The graph of is the parent graph, , shifted right units;

3. Reflections: a.) Reflection across the x -axis:

The graph of is the reflection of the parent graph, , across the x-axis.b.) Reflection across the y -axis:

The graph of is the reflection of the parent graph, , across the y-axis.

4. Vertical Stretch or Compression: The graph of is the graph of the parent graph, , … … stretched vertically by a factor of for , or

… compressed vertically by a factor of for .For , the graph is also reflected across the x-axis.

5. Horizontal Stretch or Compression: The graph of is the graph of the parent graph, , … … compressed horizontally by a factor of for , or … stretched horizontally by a factor of for .For , the graph is also reflected across the y-axis.

Vertical and HorizontalTranslations

Vertical Stretchingand Compressing

Horizontal Stretchingand Compressing

- bold - thin

- dotted

- bold - thin

- dotted

- bold - thin

- dotted

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Procedure: Multiple TransformationsTo graph , apply the transformations in the following order:

1. Horizontal translation (h)2. Reflecting/stretching/shrinking (a)3. Vertical translation (k)

Definition: Symmetric about the y -Axis If for every value of x in the domain of the function , then is called an even function and its graph is symmetric about the y-axis.

Definition: Symmetric about the OriginIf for every value of x in the domain of the function , then is called an odd function and its graph is symmetric about the origin.

Exercises:

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function.

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Translations (Shifts)

Exercises: A function g(x) along with its parent function f(x) has been given. Describe any horizontal or vertical translation from the parent function to the given function. If there is no translation, just write none.Problem #7 has already been done for you! (V.T. = Vertical translation , H.T. = Horizontal Translation.)

----------------------------------------------------------------------------------------------------------------------------------------------7) Parent function: V.T. __ Yes, 3 units up______

H.T. __Yes, 2 units to the right__ ----------------------------------------------------------------------------------------------------------------------------------------------8) Parent function: V.T. _________________________

H.T. _________________________ ----------------------------------------------------------------------------------------------------------------------------------------------9) Parent function: V.T. _________________________



H.T. _________________________ ----------------------------------------------------------------------------------------------------------------------------------------------

Sketch the following graphs of g. Use the concept of translations based off the parent functions. Also, state the domain and range.

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Section 2.3 – Families of Functions, Transformations, and SymmetryReflections

Exercises: A function g(x) along with its parent function f(x) has been given. Describe any reflection from the parent function to the given function. If there is no reflection, just write none. Problem #15 has already been done for you!

-----------------------------------------------------------------------------------------------------------------------------------------------15) Parent function: Reflection about Y-axis: None___

Reflection about X-axis: Yes____-----------------------------------------------------------------------------------------------------------------------------------------------16) Parent function: Reflection about Y-axis: _______

Reflection about X-axis: ________-----------------------------------------------------------------------------------------------------------------------------------------------17) Parent function: Reflection about Y-axis: _______



Reflection about X-axis: ________-----------------------------------------------------------------------------------------------------------------------------------------------

Sketch the following graphs of g. Use the concept of reflections based off the parent functions. Also, state the domain and range.

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Stretching & Compressing

Exercises: A function g(x) along with its parent function f(x) has been given. Describe any horizontal or vertical stretching or compression from the parent function to the given function. If there is no stretching or compressing, just write none. Problem #23 has already been done for you!

---------------------------------------------------------------------------------------------------------------------------------------------23) Parent function: Vertical: _ Yes, stretch by a factor of 2 units__

Horizontal: _____________None_____________ ---------------------------------------------------------------------------------------------------------------------------------------------24) Parent function: Vertical: _ ______________________________

Horizontal: _______________________________ ---------------------------------------------------------------------------------------------------------------------------------------------25) Parent function: Vertical: _ ______________________________



Horizontal: _______________________________ ---------------------------------------------------------------------------------------------------------------------------------------------

Sketch the following graphs of g. Use the concept of stretching and compressing based off the parent functions. Also, state the domain and range.

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Transformations (Translations, Reflections, Stretching & Compressing)

Exercises: A function g(x) along with its parent function f(x) has been given. Describe any transformation, including any vertical/horizontal translation, reflection about the x-axis/y-axis or vertical stretching or compression, from the parent function to the given function. If there are none of any of the above, just write none. Problem #31 has already been done for you!

---------------------------------------------------------------------------------------------------------------------------------------------31) Parent function:

Reflection about Y-axis: _None____ Horizontal Stretch: __ _____None_________

Reflection about X-axis: _ Yes_____ Horizontal Compression: __ _____None_________

Vertical Stretch: Yes, by a factor of 2 units__ H.T. ___Yes, 3 units to the right___

Vertical Compression: __ _____None__________ V.T. ______ Yes, 1 unit up_______

---------------------------------------------------------------------------------------------------------------------------------------------32) Parent function:

Reflection about Y-axis: __________ Horizontal Stretch: __ ___________________

Reflection about X-axis: _ _________ Horizontal Compression: _ ____________________

Vertical Stretch: ________________________ H.T. ___________ _ _______________

Vertical Compression: __ ____________________ V.T. ______ ___________________

---------------------------------------------------------------------------------------------------------------------------------------------33) Parent function:

Reflection about Y-axis: __________ Horizontal Stretch: _ ____________________

Reflection about X-axis: _ _________ Horizontal Compression: _ ____________________

Vertical Stretch: ________________________ H.T. _________________ _ _________

Vertical Compression: __ ____________________ V.T. ______ ___________________

---------------------------------------------------------------------------------------------------------------------------------------------

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Exercises:

Sketch the following graphs of g. Use the concept of transformations based off the parent functions. Also, state the domain and range.

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Math 170 - Section 2.3 - TopCatMath€¦ · Web view · 2016-06-103) 4) 5) 6) Translations ......

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