Unit 2

Transformations

Notes

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Overview: In this unit, we will develop an understanding of the effects of transformations (operations which move (or map) a figure from an original position to a new position) on the graphs of functions and their related equations. The transformations we will consider are translations, reflections, stretches, and combinations of these. In particular, we will consider replacements for x and/or y in the function y=f (x ) and investigate how the function y−k=af [b ( x−h ) ]+k is related to y=f (x ).

Schedule

Day 1 Friday, September 15th Horizontal and Vertical Translations

Day 2 Monday, September 18th Quiz – Vertical and Horizontal Translations

Reflections

Day 3 Tuesday, September 19th Quiz – Reflections

Stretches

Day 4 Wednesday, September 20th Combining Transformations

Day 5 Thursday, September 21st Quiz – Combining Transformations

Review

Day 6 Monday, September 25th Unit Test

Outcomes:

RF2: Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations.RF3: Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations.RF4: Apply translations and stretches to the graphs and equations of functions.RF5: Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the:

• x-axis• y-axis• line y=x. RF6: Demonstrate an understanding of inverses of relations.

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Lesson #1 – Horizontal and Vertical Translations

Translations

A translation is a transformation which slides each point of a figure the same distance in the same direction.

Comparing the Graphs of y=f (x ) and y−k=f (x ) or y=f ( x )+k

Part 1

a) The graph of y=x2 is shown. Use your calculator to graph y−3=x2

¿ y+3=x2 . b) In the second equation, y has been replaced by y−3. What is the effect of this

replacement on the graph of y=x2?c) In the third equation, y has been replaced by y+3. What is the effect of this

replacement on the graph of y=x2?

Part 2

a) Use a graphing calculator to graph the following functions:

b) The equation y=|x|+2 can be rewritten as y−2=|x|. How does the replacement of y by y−2 affect the graph of y=|x|?

c) The equation y=|x|−3 can be rewritten as y+3=|x|. How does the replacement of y by y+3 affect the graph of y=|x|?

d) Using the results of Parts 1 and 2, answer the following questions based on the graph of y=f (x ).i. What is the effect of the parameter k on the graph of the function y=f ( x )+k?

ii. What is the effect of the parameter k on the graph of the function y−k=f ( x )?e) Complete the following statements:

Compared to the graph of y=f ( x ), the graph of y−k=f ( x ) results in a _________________ translation of k units.

If k>0, then the graph moves ___________. If k<0, then the graph moves __________.

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Note: The notation y−k=f ( x ) is often used instead of y=f ( x )+k to emphasize that this is a transformation on y. The concept of replacing y by y−k will be very important in this course.

Comparing the Graphs of y=f (x ) and y=f (x−h)

Part 1

a) The graph of y=x2 is shown. Use your calculator to graph y= (x−3 )2∧ y=(x+3)2.

b) In the second function, x has been replaced by x−3. What is the effect of this replacement on the graph of y=x2?

c) In the third function, x has been replaced by x+3. What is the effect of this replacement on the graph of y=x2?

Part 2

a) Use a graphic calculator to graph the following functions, observing the effects of replacing x by x−4, and x byx+2.

i. y=√x ii. y=√x−4 iii. y=√x+2

b) Based on the graph of y=f ( x ), and using the results of Parts 1 and 2, describe the effect of the parameter h on the graph of the function y=f ( x−h ) .

c) Complete the following statements: Compared to the graph of y=f ( x ), the graph of y=f ( x−h ) results in a _________________

translation of h units. If h>0, then the graph moves ___________. If h<0, then the graph moves __________.

Replacements for Translations

Given the function ¿ f (x) :

replacing y with y−k (i.e. y y−k) describes a vertical translation, y−k=f (x ) or y=f ( x )+k describes a vertical translation

replacing x with x−h (i.e. x x−h) describes a horizontal translation, y=f (x−h) describes a horizontal translation

In general, if

y−k=f (x−h)

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k>0the graph moves up ↑

k<0 the graph moves down ↓

h>0 the graph moves right →

h<0 the graph moves left ←

or

y=f ( x−h )+k then

Example #1: What happens to the graph of the function y=f (x ) if the following changes are made to its equation?

a) replace x with x+2 b) replace y with y−8

Example #2: Describe how the graphs of the following functions relate to the graph of y=f (x ).

a) y=f ( x−3 ) b) y=f ( x )+4 c) y−1=f (x+ 10)

Example #3: The point (2 ,−3) lies on the graph of y=f (x ). State the coordinates of the image of this point under the following transformations.

a) y+8=f ( x ) b) y=f ( x−7 )+5

Example #4: Write the equation of the image of y=f (x ) under each transformation.

a) A horizontal translation of 5 units left.

b) A translation of 3 units up.

c) A translation of m units right and p units down.

Example #5: Given the graph of the function y=f (x ), sketch the graph of the indicated function.

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Replacement Notation and Mapping Notation

Do not confuse mapping notation with the notation we have used for replacements.

Consider the example where the graph of y=f ( x ) is transformed to the graph y−2=f ( x−3 ) .

In this example, the replacements for x and y may be written as x→ x−3 and y → y−2.

Under this transformation, all points on the graph of y=f (x ) will move 3 units to the right and 2 units up. The point with coordinates (4,6) will be translated to the point (7,8) . In general the point with coordinates (x , y ) is translated to the point (x+3 , y+2).

The mapping notation for this translation may be written as (x , y )→ (x+3 , y+2), implying that the point with coordinates (x , y ) is translated to the point (x+3 , y+2).

Notice that the mapping notation (x , y )→ (x+3 , y+2) is NOT the same as the replacement notation x→ x−3 and y → y−2.

The mapping notation (x , y )→ (x+3 , y+2) is equivalent to the replacement notation x→ x−3 and y → y−2.

Example #6:

a) State the coordinates of the image of the point (−3,5) under the translation described by (x , y )→ (x−7 , y+4).

b) Write the equation of the image of y=f ( x ) after the translation (x , y )→ (x−6 , y+1)

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Example #7: Write the replacements for x and/or y, and describe how the graph of the second function compares to the graph of the first function.

a) y=x4 , y=x 4+3

b) y=6 x−3 , y=6 ( x−1 )−3

c) y=|x|, y=|x−6|+2

d) y= 1√ x

, y= 1√x+1

Example #8: Write the replacements for x and/or y, and the equation of the image of

a) y=x2 after a horizontal translation of 3 units to the right.

b) y=10x after a vertical translation of 2 units up

c) y=√x after a horizontal translation of 4 units to the left and a vertical translation of 3 units down.

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Example #9: The function represented by the thick line is a transformation of the function represented by the thin line. Write an equation for each function represented by the thick line.

Example #10:y=√xis radical function

a) What vertical translation would be applied to y=√x so that the translation image passes through (16 , 7 )?

b) What horizontal translation would be applied to y=√x so that the translation image passes though (17,8 )?

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Lesson #2 – Reflections

Invariant points are points on a graph which do not move after a transformation.

Comparing the Graphs of y=f (x ) and y=−f (x )

Part 1

a) The graph of y=f ( x )=x2−10 x+25is shown. Write an equation

which represents y=−f (x ).

b) Use a graphing calculator to sketch y=−f (x ) and show the graph on the grid.

c) State the coordinates of the invariant point(s).

Part 2

a) The graph of y=f ( x )=x3−8 is shown. Write an equation which

represents y=−f (x ).

b) Use a graphing calculator to sketch y=−f (x ) and show the graph on the grid.

c) State the coordinates of the invariant point(s).

d) How does the graph of y=−f (x ) compare with the graph of y=f ( x ).

Part 3

The graph of y=f (x ) is shown. Sketch the graph of y=−f (x ).

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Note: If we replace y with – y, then y=f (x ) becomes − y=f (x ), which is equivalent to y=−f (x ). So the replacement in this example is y →− y.

Comparing the Graphs of y=f (x ) and y=f (−x )

Part 1

a) The graph of y=f ( x )=x2−10 x+25is shown. Write an

equation which represents y=f (−x ).

b) Use a graphing calculator to sketch y=f (−x ) and show the graph on the grid.

c) State the coordinates of the invariant point(s).

Part 2

a) The graph of y=f ( x )=x3−8 is shown. Write an equation

which represents y=f (−x ).

b) Use a graphing calculator to sketch y=f (−x ) and show the graph on the grid.

c) State the coordinates of the invariant point(s).

d) How does the graph of y=f (−x ) compare with the graph of y=f ( x ).

Part 3

The graph of y=f (x ) is shown. Sketch the graph of y=f (−x ).

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Note: If we replace x with −x, then y=f (x ) becomes y=f (−x ). So the replacement in this example is x→−x.

Comparing the Graphs of y=f (x ) and y=f −1(x ) and x=f ( y )

Part 1

a) The graph of y=f ( x )=(x−5)2is shown. Write an equation

which represents x= f ( y )

b) Use a graphing calculator to sketch x=f ( y ) and show the graph on the grid.

c) State the coordinates of the invariant point(s).

Part 2

a) The graph of y=f ( x )=x3−8 is shown. Write an equation

which represents ¿ f−1(x) .

b) Use a graphing calculator to sketch y=f −1(x ) and show the graph on the grid.

c) State the coordinates of the invariant point(s).

d) How does the graph of y=f −1(x ) compare with the graph of y=f ( x ).

Part 3

The graph of y=f (x ) is shown. Sketch the graph of x=f ( y ).

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Note: If we replace x with y, and y with x, then y=f (x ) becomes x= f ( y ). So the replacement in this example are x → y and y→ x.

Reflections

A reflection is a transformation which reflections (or flips) a figure about a line. Fill in the following blanks which summarize the previous investigations.

Note: Given the function y=f (x ):

replacing x with – x, (ie. x→−x) describes a reflection in the y-axis, y=f (−x ) describes a reflection in the y-axis

replacing y with – y, (ie. y →− y) describes a reflection in the x-axis, − y=f (x ) or y=−f (x )describes a reflection in the x-axis

interchanging x and y, (ie. x → y and y→ x) describes a reflection in the line y=x , x=f ( y)or y=f −1(x ) describes a reflection in the line y=x

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Example #1: The graph of y=f (x ) is shown. Sketch:

Example #2: The graph drawn in the thick line is a reflection of the graph drawn in the thin line. Write an equation for each graph drawn in the thick line.

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Combining Reflections Part 1 – Transforming y=f (x ) to y=−f (−x)The table below shows how to “build” y=−f (−x) from y=f (x )

The transformations in the table are applied to shape A in the order shown. The images are shown on the grid.

a) Complete the table below to determine the equation which results from changing the order in which the reflections are carried out.

b) On the grid above, sketch the image of the shape A under the combinations of transformations in a).c) Does the order in which the reflections are carried out affect the final image?

d) Describe two sets of transformations, in order, which can be applied to the graph of y=f ( x ) to produce the graph of y=−f (−x ) .

Part 2 – Transforming y=f (x ) to x=f (− y ) and x=−f ( y )

In each case, complete the table and sketch the combination of transformations on the grid.

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a) Describe two sets of transformations, in order, which can be applied to the graph of y=f (x ) to produce the graph of x= f (− y ).

b) Describe two sets of transformations, in order, which can be applied to the graph of y=f (x ) to produce the graph of x=−f ( y ).

Example #3: Write the equation of the image of:

a) y=x2 after a reflection in the line y=x

b) y=10x after a reflection in the y-axis

c) y=√x after a reflection in the x-axis

Example #4: Describe how the graph of the second function compares to the graph of the first function.

Example #5: The graph drawn in the thick line is a transformation of the graph drawn in the thin line. Write an equation for each graph drawn in the thick line and state whether this graph represents a function.

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Example #6:

a) Sketch the graph of f ( x )= 6x2+3

.

b) Write the equation for

i. y=−f (x )

ii. y=f (−x ) iii. x=f ( y )

c) Sketch each graph in b) and state whether the graph represents a function.

Example #7: a) Given f (x)=3x+2, determine:

i. x=f ( y ) ii. x=f (− y ) iii. x=−f ( y )

b) The graph of y=3 x+2 is given. Sketch each graph in a).

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Lesson #3 – Stretches About the x or y-axisComparing the Graphs of y=f (x ) and y=af (x), where a>0

Part 1

The graph of y=f (x )=√4−x2 is shown.

a) Write an equation which represents y=3 f (x ).

b) Use a graphing calculator to sketch y=3 f (x ) on the grid. c) Describe how the number 3 in y=3 f (x ) affects:

i. the general sketch of y= f ( x )

ii. the x-intercepts of the graph of y=f ( x ) .

iii. the y-intercept of the graph of y=f ( x ) .

Part 2

a) Write an equation which represents y=12

f (x ).

b) Use a graphing calculator to sketch y=12

f (x ) on the grid.

c) Describe how the number 12 in y=

12

f (x ) affects:

i. the general sketch of y= f ( x )

ii. the x-intercepts of the graph of y=f ( x ) .

iii. the y-intercept of the graph of y=f ( x ) .

d) Complete the following statement using the results of Parts 1 and 2. Compared to the graph of y=f (x ), the graph of y=af (x) results in a __________________ stretch about the ___-axis by a factor of ____.

Compared to the graph of y=f (x ), the graph of y=af (x), where a<0, results in a _____________ stretch about the _____-axis by a factor of |a| together with a reflection in the _______-axis.

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Note: If we replace y with 1a

y , then y= f ( x ) becomes 1a

y=f (x), which is equivalent to y=af (x). So the

replacement in this example is y → 1a

y.

Comparing the Graphs of y=f (x ) and y=f (bx ), where b>0

Part 1

The graph of y=f (x )=√4−x2 is shown.

a) Write an equation which represents y=f (4 x) .

b) Use a graphing calculator to sketch y=f (4 x) on the grid. c) Describe how the number 4 in y=f (4 x) affects:

i) the general sketch of y=f ( x )

ii) the x-intercepts of the graph of y=f ( x ) .

iii) the y-intercept of the graph of y=f ( x ).

Part 2

a) Write an equation which represents y=13

f (x ).

b) Use a graphing calculator to sketch y=13

f (x ) on the grid.

c) Describe how the number 13 in y=1

3f (x ) affects:

i. the general sketch of y= f ( x )

ii. the x-intercepts of the graph of y=f ( x ) .

iii. the y-intercept of the graph of y=f ( x ) .

d) Complete the following statement using the results of Parts 1 and 2. Compared to the graph of y=f (x ), the graph of y=f (bx ) results in a __________________ stretch about the ___-axis by a factor of ____.

Compared to the graph of y=f (x ), the graph of y=f (bx ), where b<0, results in a _____________

stretch about the _____-axis by a factor of 1|b| together with a reflection in the _______-axis.

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Note: If we replace x with bx, then y=f ( x ) becomes y=f (bx ). So the replacement in this example is x→ bx.

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StretchesIn mathematics we use the word stretch to represent both an expansion and a compression. In this course we only consider stretches about the x- and y-axis.

In the table below, the graph of y=f ( x ) and the graph of y=af (x) or y=f (bx ) is given. Fill in the blanks.

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Given the function y= f ( x ): replacing x with bx, (ie. x→ bx) describes a horizontal stretch about the y-axis, ie. y=f (bx ) describes a

horizontal stretch

replacing y with 1a

y , (ie. y → 1a

y) describes a vertical stretch about the x-axis, ie. 1a

y=f ( x ) or

y=af (x) describes a vertical stretch

In general, if 1a

y=f (bx) or y=af (bx ), then for

a>0 – vertical stretch about the x-axis by a factor of a

a>0 – vertical stretch about the x-axis by a factor of |a| and a reflection in the x-axis

b>0 – horizontal stretch about the y-axis by a factor of 1b

b<0 – horizontal stretch about the y-axis by a factor of 1|b| and a reflection in the y-axis

Example #1: Write the replacement for x or y and write the equation of the image of y=f ( x ) after each transformation.

a) a horizontal stretch by a factor of 6 about the y-axis

b) a vertical stretch by a factor of 15 about the x-axis

c) a reflection in the x-axis and a vertical stretch about the x-axis by a factor of 3

d) a horizontal stretch about the y-axis by a factor of 12 and a vertical stretch about the x-axis by a factor of

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Example #2: How does the graph of 3 y=f (x ) compare with the graph of y=f (x )?

Example #3: What happens to the graph of the function y=f (x ) if you make these changes?

a) Replace x with 4 x. b) Replace y with 13

y. c) Replace y with 6 y and x with 13

x

Example #4: The graph of y=f (x ) is shown. Sketch y=f (−2 x).

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Example #5: Write the equation of the image of

a) y=x2 after a horizontal stretch about the y-axis by a factor of 34

b) y=√x−3 after a horizontal stretch by a factor of 4 about the y-axis and a vertical stretch by a factor of 2 about the x-axis

c) y=3 x+7 after a vertical about the x-axis by a factor of 13 and a reflection in the x-axis

Example #6: Describe how the graph of the second function compares to the graph of the first function.

Example #7: The function represented by the thick line is a stretch of the function represented by the thin line. Write an equation for each function represented by the thick line.

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Example #8: A polynomial function has the equation P(x )=(x−4)(x+3)(x+6). Determine the zeros and the y-intercept if the following transformations are applied.

a) y=−3P ( x )b) y=P(−1

2x)

Lesson #4 – Combining Transformations

Combining Transformations

In the previous lessons, we have learned the following rules:

Given the equation of a function y=f (x )

replacing x with x−h or y with y−k results in a horizontal or vertical translation replacing x with – x or y with – y results in a reflection

replacing x with bx or y with 1a

y results in a horizontal or vertical stretch

Investigation #1: Combining a Horizontal Stretch and a Vertical Stretch

The graph of y=f (x ) is shown.

a) Sketch the image of the function after a horizontal stretch by a factor of 3 about the y-axis, followed by a vertical stretch by a factor of 2 about the x-axis.

b) Sketch the image of the function after a vertical stretch by a factor of 2 about the x-axis, followed by a horizontal stretch by a factor of 3 about the y-axis.

c) Does the order in which the two stretches are performed make a difference to the final graph?

Investigation #2: Combining a Vertical Translation and a Horizontal Stretch

The graph of y=f (x ) is shown.

a) Sketch the image of the function after a vertical translation of 3 units up, followed by a horizontal stretch by a factor of 2 about the y-axis.

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b) Sketch the image of the function after a horizontal stretch by a factor of 2 about the y-axis, followed by a vertical translation of 3 units up.

c) Does the order in which the vertical translation and the horizontal stretch are performed make a difference to the final graph?

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Investigation #3: Combining a Vertical Translation and a Vertical Stretch

The graph of y=f (x ) is shown.

a) Sketch the image of the function after a vertical translation of 2 units down, followed by a vertical stretch by a factor of 3 about the x-axis.

b) Sketch the image of the function after a vertical stretch by a factor of 3 about the x-axis, followed by a vertical translation of 2 units down.

c) Does the order in which the vertical translation and the vertical stretch are performed make a difference to the final graph?

Investigation #4: Combining a Horizontal Translation and a Horizontal Stretch

The graph of y=f (x ) is shown.

a) Sketch the image of the function after a horizontal translation of 1 unit right, followed by a horizontal stretch by a factor of 2 about the y-axis.

b) Sketch the image of the function after a horizontal stretch by a factor of 2 about the y-axis, followed by a horizontal translation of 1 unit right.

c) Does the order in which the horizontal translation and the horizontal stretch are performed make a difference to the final graph?

Order of Transformations

We have seen that when two transformations are applied to a graph, the order in which the transformations are performed may or may not make a difference to the final graph.

In general, the order DOES NOT matter when:

two translations are combined two stretches are combined

a translation and a stretch at right angles to one another are combined

reflections and stretches are combined

The order DOES matter when:

a translation and a stretch in the same direction are combined most reflections and translations are combined

Note: Unless otherwise indicated, use the following order to describe how to transform from one graph to another:

1. Stretches 2. Reflections 3. Translations 26

Example #1: Describe a series of transformations required to transform graph A to graph B.

Example #2: Describe a series of transformations required to transform:

a) graph A to graph B

b) graph A to graph C

c) graph B to graph C

Example #3: Describe which transformations are applied to a graph of a function when the following changes are made to its equation. Does the order in which the transformations are performed affect the final graph?

a) Replace x with 3 xand y with y+4b) Replace x with

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x, y with −3 y, and x with

x+2

Example #4: A graph of the parabola y=x2+1 is shown. The following

transformations are applied to y=x2+1 in the order shown.

a horizontal translation 2 units left a reflection in the x-axis a vertical stretch about the x-axis by a factor of 0.5 a vertical translation 3 units down

a) For each transformation: graph the image on the grid write the replacement for

x or y and the current equation in the table

b) Write the equation which represents the final position of the graph and verify using a graphing calculator

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Equations Combining Two or More Transformations

To apply a combination of transformations, consider the following:

y=af [b (x−h ) ]+k where

|a|is the vertical stretch factor. If a is negative, there is also a reflection in the x-axis.

1|b| is the horizontal stretch factor. If b is negative, there is also a reflection in the y-axis.

h is the horizontal translation where

if h>0, the translation is to the right ifh<0, the translation is to the left

k is the vertical translation where

if k>0, the translation is k units up ifk<0, the translation is k units down

Note: When graphing a combination of transformations from an equation, use the following order:

Step 1: Sketch the original functionStep 2: Sketch any stretchesStep 3: Sketch any reflectionsStep 4: Sketch any translations

Example #5: The graph of y=f (x ) is shown.

Consider the function defined by the equation

y=2 f ( 12

( x+5 ))−8.

a) If the equation is written in the form y=af [b (x−h ) ]+k ,

state the values of a , b ,h, and k .

b) Write the transformations associated with the parameters a , b , h, and k .

c) Put these transformations in an order which can be used to sketch the graph of the function. Sketch the graph of the function.

d) The original graph has equation y=x2. Write the equation for the transformed function

y=2 f ( 12

( x+5 ))−8.

e) Graph the equation in a) on a calculator and verify the sketch c).

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Example #6: The graph of y=f (x ) is shown.

Sketch the graph of y=−2 f (x−3 )+1.

Example #7: Consider the function y=f (x ). In each case determine:

the replacements for x and y which would result in the following combinations of transformations the equation of the transformed function in the form y=af [b (x−h ) ]+k

a) a horizontal stretch by a factor of 14 about the y-axis and a vertical translation of 5 units down

b) a vertical stretch by a factor of 35 about the x-axis, a reflection in the y-axis, and a horizontal translation of

2 units left

Example #8: A function G ( x )=x3 is transformed into a new function P(x ). To form the new function P ( x ) , G(x ) is stretched vertically about the x-axis by a factor of 0.25, reflected in the y-axis, and translated 3 units to the right. Write the equation of the new function P(x ).

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Example #9: Given the graph of y=f (x ), sketch the graph of the transformed function y=f ( 12

x+3)−8.

Example #10: The function f ( x )=√x has ben transformed in to the function g ( x )=−2√3 x−12+5. Complete the following statement.

The function f(x) has been transformed to the function g(x) by stretching horizontally about eh y-axis by a factor of ______, stretching vertically about the x-axis by a factor of _____, reflecting in the _________, translating ______ units up and ______ units horizontally to the _________.

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