1.1 horizontal & vertical translations - · pdf fileunit ii transformations of functions 1...

Unit II Transformations of Functions 1

1.1 Horizontal & Vertical Translations

(I) Determining the effects of k in y - k = f(x) or y = f(x) + k

on the graph of y = f(x)

Example: Given the base function y = f(x)

(a) Use a table of values for each indicated function to produce a graph

on the coordinate grid.

Function Table of Values Graph

Base Function

f(x) = x2

x f(x) = x2

Goal: Demonstrate an understanding of the effects of horizontal and

vertical translations on the graphs of functions and their related

equations.

Determining the effects of h and k in y – k = f(x – h) on the

graph of y = f(x)

Sketching the graph of y – k = f(x – h) for given values of

h and k, given the graph of y = f(x)

Writing the equation of a function whose graph is a vertical

and/or horizontal translation of the graph of y = f(x)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345



y = f(x) + 2

x y=f(x)+2

y – 2 = f(x)

x y-2=f(x)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-2

-1

1

2

3

4

5

6

7

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-2

-1

1

2

3

4

5

6

7

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-2

-1

1

2

3

4

5

6

7

General Rule about the effect of k

What impact does the value of k for y = f(x) + k or y – k = f(x) have on the

transformation of the base graph y = f(x)?

The value of k represents a _________________

This _________________ affects the ________ but not the __________ or

_________________


Sketching the graph of y – k = f(x) or y = f(x) + k given a base graph

y = f(x)

Example:

Given the base graph:

(a) Identify key points for y= f(x) and create a table of values.

x y = f(x)

(b) Create a new table of values for y + 2 = f(x) and

Sketch the graph on the grid above.

x y + 2 = f(x)

Image Points

The point that is the result of a

transformation of a point on the

original graph.

next

to each letter representing an

image point.


(II) Determining the effects of h in y = f(x – h) on the graph

of y = f(x)

Example: Given the base function y = f(x)

(a) Use a table of values for each indicated function to produce a graph

on the coordinate grid.


Base Function

f(x) = |x|

x f(x) = |x|

Mapping Notation

Each point (x, y) on the base graph of y = f(x) in (a) above is transformed in

(b) to become the point (x, ) on the graph of y + 2 = f(x).

Using mapping notation (x, y) → (x, )

Mapping

Relating one set of points to another set of points so each point in the original

set corresponds to exactly one point in the image set.

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345



y = f(x – 1)

y = f(x + 2)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

General Rule about the effect of h

What impact does the value of h for y = f(x – h) have on the transformation of the base

graph y = f(x)?

The value of h represents a _________________

This _________________ effects the ________ but not the __________ or

_________________

Mapping rule (x, y) → ( , y)


Sketching the graph of y = f(x – h) given a base graph y = f(x)

Example: Given the base graph y = f(x)

Create a mapping rule

a table of values

and sketch the graph on the grid above for y = f(x + 3)

Mapping Rule: (x, y) → ( , )

x- 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

y

- 2

- 1

1

2

3

4

5

6

7

8

x- 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

y

- 2

- 1

1

2

3

4

5

6

7

8


(III) Describing the translation of each function when compared

to y = f(x).

Example:

Express the mapping rule and describe the translation of each

function compared to y = f(x).

(a) y = f(x – 2)

(b) y = f(x) – 9

(c) y = f(x + 3) – 7

(d) y – 12 = f(x + 4)


(IV) Sketching the graph of y – k = f(x – h) or y = f(x – h) + k given a

base graph of y = f(x).

Example:

For each function:

(i) State the mapping rule

(ii) Create a table of values

(iii) Graph the transformed functions

(a) y + 2 = f(x – 6)

(x, y) ( , )

(b) y = f(x + 2) + 5

(x, y) ( , )


(V) Writing the equation of a function based on a transformation

of the base function y =f(x)

Example:

Determine the values of h and k and write the equation of the translated

graph.

P.12 – 15 #2 - #4,#8, #11,C1


1.2 Reflections and Stretches

(I) Graphing Reflections in the x and y-axis

Reflections in the x-axis:

Consider the point A(2, 3) and

plot it in the coordinate grid.

If the x-axis represents a mirror

(or reflection line), then plot and

state the coordinates of the image

point A⁄.

Coordinates of A⁄ __________

Mapping a reflected point in the x-axis:

Mapping the point A to A⁄ is represented by

A → A⁄

(x, y) → ( )

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

Goal:

Developing an understanding of the effects of reflections on the

graphs of functions and their related equations


Reflections in the y-axis:

Consider the point A(2, 3) and

plot it in the coordinate grid.

If the y-axis represents a mirror

(or reflection line), then plot and

state the coordinates of the image

point A⁄.

Coordinates of A⁄ __________

Effects of Reflections on Graphs and Equations

Given the graph of y = f(x),

sketch the graph of y = –f(x)

using a mapping rule or transformations.

Mapping a reflected point in the y-axis:

Mapping the point A to A⁄ is represented by

A → A⁄

(x, y) → ( )

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

In general y = –f(x) represents a ____________ in the __ axis.


Effects of Reflections on Graphs and Equations

Given the graph of y = f(x),

sketch the graph of y = f(–x)

using a mapping rule or transformations.

In general y = f(–x) represents a ____________ in the __ axis.

Summary of Reflections Creates a mirror image through a reflection line

Does NOT change the _________ of the graph

DOES change the _____________ of the graph


Equation of a Function from a Graph & Invariant points

In each graph below the function y = f(x) and a transformed graph is provided. In

each case:

(a) State the type of transformation

(b) The mapping rule

(c) The equation of the transformed function

(d) Identify any invariant points.

1.

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

y = f(x)

Remember: Invariant Points

A point that remains unchanged wrt position after a transformation is applied

A point on a curve that lies on the line of reflection

Type of transformation _____________

Mapping rule (x, y) → (

Equation: __________________

Invariant points:


2.

3. Which of the following transformations would produce a graph with the

same x-intercepts as y = f (x)?

(A) y = – f (x) (B) y = f (–x) (C) y = f (x + 1) (D) y = f (x) + 1

4. Which axis was the first point reflected through to get

the coordinates of the second point?

(i) (6, 7) and (−6, 7) (ii) (−2, −7) and (−2, 7)

Type of transformation _____________

Mapping rule (x, y) → (

Equation: __________________

Invariant points:

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345


(II) Graphing Vertical and Horizontal Stretches

Graphing Vertical Stretches:

Plot the point A(1, 2)

in the coordinate grid.

Plot a point A⁄ with the

same x-coordinate as A

and a y-coordinate 2 times

the coordinate in A.

Plot a point A⁄⁄ with the

same x-coordinate as A

and a y-coordinate

times

the coordinate in A.

Describe how multiplying the y-coordinate by a factor of 2 or by a factor of

affects the position of the image point.

Mapping a vertically stretched point:

Mapping the point A to A⁄ or A

⁄⁄ is represented by

A → A⁄ or A

⁄⁄

(x, y) → ( )

x-1 1 2 3 4 5

y

-1

1

2

3

4

5


Effects of Vertically Stretching on Graphs and Equations

Given the base graph of y = f(x) identify the key points

x y = f(x)

(a) Generate a table of values to produce the graph of y = 2f(x) or

y = f(x).

(b) Generate a table of values to produce the graph of y =

f(x) or 2y = f(x).

x y = 2f(x)

x y =

f(x)


Reflection and Stretching

Given the graph of y = f(x) and y = af(x) or

determine:

(a) the vertical stretch

(b) whether the vertical stretch

can ever be negative.

(c) the mapping rule.

(x, y) ( )

(d) the equation of the function.

(e) what effects will be on the function when the:

(i) |a| > 1

(ii) |a| < 1

In general

y = af(x) or

represents a _________________

The value of ‘a’ changes the _________ of the graph

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

y=f(x)


Effects of Horizontally Stretching on Graphs and Equations

Example:

Below is the base graph y = f(x) [or y = sin(x)]

Below is the graph of y = f(2x) [or y = sin (2x)].

Produce the table of values and state the mapping rule.

Mapping Rule: (x, y) ( )

x- 90 ° 90 ° 180 °

y

- 2

- 1.5

- 1

- 0.5

0.5

1

1.5

2


Example:

Below is the graph of y = f(

x) [or y = sin (

x)].



Example:

Below is the graph of y = f(–2x) [or y = sin (–2x)].



x- 90 ° 90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 2

- 1.5

- 1

- 0.5

0.5

1

1.5

2

x- 90 ° 90 ° 180 °

y

- 2

- 1.5

- 1

- 0.5

0.5

1

1.5

2


Effects of Vertical/Horizontal Stretching on Graphs and Equations

When we have both vertical and horizontal stretching on the base

base graph y = f(x) we have to consider the effect of ‘a’ and ‘b’

on the graph of y = af(bx) or

Example: Given the graph of y = f(x) and y = af(bx)

determine the span of each domain and range and

write the equation of the transformed graph

General effects of a horizontal stretch on a base graph

A horizontal stretch is always _________________

Given the function y = f(bx), the mapping rule is

(x, y) → ( )

If b < 0, the graph will be _____________ as well as reflected in the ____ axis.

HS =

P.28 – 31 #3 - #10, #14, #15, C2, C3, C4


1.3 Combining Transformations

(I) Sketch the graph of a function y – k = af(b(x – h)) for given

values of a, b, h and k given the graph of the function y = f(x)

Example:

Describing the transformations of the function y = f(x) based on the

transformed function y = –2f(3(x – 1)) + 4.

Horizontal stretch of _____

Vertical stretch of _____

Reflection in the ______

Horizontal translation ____________

Vertical translation ___________

Transformations:

(I) Stretches and Reflections are the result of _________________

(II) Horizontal/Vertical Translations are the result of ___________

Due to the importance of the order of operations, ________________

are applied first.

Goal:

Sketching the graph of a transformed function by applying

translations, reflections and stretches


Example:

Using the graph of the function y = f(x), graph the transformed

function y = –2f(3(x – 1)) + 4

Create a table of values for the transformed function and graph the function

on the grid above.

x y = f(x)

Transformed graph y = –2f(3(x – 1)) + 4

Determine the mapping rule

for y = f(x) based on the

transformed function

y = –2f(3(x – 1)) + 4.

(x, y) →

Create a table of values for

the base graph of y= f(x).


Note:

It is sometimes necessary to rewrite a function before it can be graphed since the

horizontal translation value can be correctly identified.

Example:

Express the mapping rule for y – 6 = 3f(4x – 8) as a transformation of y = f(x).

mapping rule: (x, y) → (

To accurately sketch the graph of a function of the form

y – k = af(b(x – h) + k.

Stretches and reflections (a and b values) should occur before translation

values (h and k values)


Example:

Given the graph of y = f(x), sketch the graph of y + 2 = 2f(–3x – 3) on the same

grid.


(II) Write the equation of a function given its graph is a translation

and/or stretch of the graph of the function y = f(x)

Example:

Compare the base graph f(x) to the graph of the transformed function g(x) to

identify all transformations and state the equation of the transformed

function.

Step I Determine the horizontal stretch (HS) and the vertical stretch (VS) of y = g(x)

by comparing the domains and ranges of y = f(x) to y = g(x).

Domain of y = f(x) ______

HS of y = g(x) ______

Domain of y = g(x) ______

Range of y = f(x) ______

VS of y = g(x) ______

Range of y = g(x) ______


Step II Consider whether the points are reflected through either the x or y axis.

Analyze the orientation of image points for y = g(x) wrt the x and y axes

compared to the position of corresponding base points on the graph of y = f(x).

Reflection in x-axis: _____ Reflection in y-axis: _____

Step III Develop a mapping rule on the basis of results for stretches and reflections in

steps I and steps II.

(x, y) ( )

Step IV Test the mapping rule from step III by taking the coordinates of one base point

from y = f(x) and determining the corresponding image coordinates for y =

g(x).

Use base point ( ) apply mapping rule (x, y) ( )

to determine corresponding image point ( ).

Step V Plot the corresponding image point for y = g(x) from step IV on the grid below

and analyze its placement to determine the appropriate horizontal and vertical

translations so that the image point will be translated to the correct position.

HT = _____

VT = _____

Step VI Apply the results for HT and VT to complete the mapping rule then write the

function y = af(b(x – h)) + k

(x, y) ( ) and ________________________________


Example:

Determine the specific equation for the image of y = f(x) in the

form y = af(b(x – h)) + k.


P. 38 – 42 #2 – #10, #12, #15, C4


1.4 Inverse of a Relation

(I) What is an Inverse Relation?

Ex. Describe the distance and direction required to travel the

route indicated below.

(a) From A to B (b) From B to A

Distance and direction Distance and direction

A to B B to A

A

B

3 km

2 km

A

B

3 km

2 km

Goals:

Defining an inverse relation

Determining the equation of an inverse

Sketching the graph of an inverse relation

Determining if a relation and its inverse are functions


NOTE: An inverse relation accomplishes 2 things:

(i) the ORDER of execution and

(ii) the OPERATION

With respect to Mathematical Relations, inverse relations have:

(i) a change in the ORDER of algebraic execution and

(ii) the algebraic OPERATION changes (inverse operation)

(II) Determining the equation of an inverse algebraically

Procedure to attain an inverse algebraically:

•change f(x) to y

•switch x and y

•solve for y

Ex. Determine the inverse function for f(x) = 2x – 3


Describe the ORDER and algebraic OPERATION on x

in each function below.

f(x) = 2x – 3

• •

• •

These functions are therefore _____________________

If f(x) = 2x + 3 then express the inverse function using inverse function

notation f –1

(x).

Note:

An inverse function can be expressed

using inverse function notation f –1

(x)


(III) Sketching the graph of an inverse relation

Example:

For each function f(x) = 2x + 3 and

(a) create a table of values

Points on f(x) = 2x + 3 Points on

–2

–1

0

1

2

(b) graph each function on the same grid.

Remember attaining an algebraic inverse:

x and y interchanges

To attain an inverse graphically

x and y will interchange

Example:

If (0, –3) lies on the graph of f(x) then what inverse

coordinates lie on the graph of f –1

(x)? _________

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 8- 7- 6- 5- 4- 3- 2- 1

12345

Sketch the reflection line

What is the equation of the

reflection line? _________

Points on the graph of f(x) are

related to the points on f –1

(x) by

the mapping rule (x, y) → (

)


Observations from graph above:

(i) What point(s) above are invariant after the reflection?

(ii) Where are these invariant points located?

(iii) If the y – intercept of a base relation is (0, b) then the inverse coordinates for

inverse relation are ( ) which represents

an _____________.

(iv) If points on the graph of the base relation are located in the:

First quadrant then the inverse coordinates are in the __________

(ie. If (a, b) is in the first quadrant then ( , ) is in the

the ____________)

Third quadrant then the inverse coordinates are in the _________

(ie. If (–a, –b) is in the third quadrant then ( , ) is in the

the ____________)

Second quadrant then the inverse coordinates are in the ________

(ie. If (–a, b) is in the second quadrant then ( , ) is in the

the ____________)


Example: Given the graph:

(a) Create a table of values

using key points.

(b) Create inverse coordinates

(c) Graph the inverse relation.

x y = f(x)

How to graphically sketch an inverse relation given a graph

(I) Method I – Create inverse coordinates from the base graph

Identify key points from the base relation and interchange the values of x and y and

plot the resulting inverse coordinates

(II) Method II – Use a blank piece of paper to create a reflection through y = x

Trace the graph including the x and y axes on a piece of paper

Flip the traced graph onto the original graph so x and y axes are lined up

Rotate the blank paper 90° so the y-axis is on the x-axis

The inverse relation will appear as an image on the underside of the blank paper.


(IV) Determining if a relation and its inverse are a function

Example: Which relation represents a function?

(A) (B)

Example: Given the relation, sketch the inverse relation by reflecting through

the line y = x or by applying the mapping rule (x, y) → (y, x)

Remember: Graphically distinguishing functions

Graphically distinguishing a function to determine a one to one

correspondence between domain (x–values) and range (y–values) by the

__________________test.

x

y

x

y

Is the inverse relation a

function?

What kind of line test

could be used on the base

graph y = f(x) to determine

if the inverse would be a

function?


Example:

Without sketching, which relation would produce an inverse function?

(V) Restricting the domain of a relation to attain an inverse function

Graphically attaining an inverse function

Example:

How can we graphically restrict the domain of the base graph

of y = 2x2 + 1 so that the inverse is a function?

Algebraically

Remember:

The ________________

predicts whether the

inverse relation would be a

function.

How much of the domain

from the given graph could

be reflected to produce an

inverse function?


Attaining an inverse function for a quadratic

Example:

Algebraically attain the inverse function (ie. f –1

(x) ) for:

(a) f(x) = 2x2 + 1.

(b) f(x) = 2(x + 1)2 + 3

State the vertex_____

Determine the y-intercept_____

Sketch the graph of f(x).

Attain the inverse function

f –1

(x) and sketch.

Procedure to attain an

inverse algebraically:

•change f(x) to y

•switch x and y

•solve for y

x- 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8

y

- 4

- 3

- 2

- 1

1

2

3

4

5

6

7

8


How to attain an inverse function for a quadratic in standard form

Example:

Determine the inverse function for y = 2x2 + 4x + 5.

Express y = 2x2 + 4x + 5 in vertex form y = (x – h)

2 + k and sketch the graph.

Restrict the domain of y = f(x) so that y = f –1

(x) is a function. Sketch the

inverse function on the same grid.

State the:

(i) Restricted domain and range for y = f(x)

Domain:__________ Range:__________

(ii) Domain and range for y = f –1

(x)

Domain:__________ Range:__________

x- 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8

y

- 4

- 3

- 2

- 1

1

2

3

4

5

6

7

8


x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

(VI) Determining algebraically or graphically if two functions

are inverses of each other

Graphically

Determine if there is symmetry about the line y = x

Example:

Which graph shows a function and its inverse?

(A) (B)

(C) (D)


Algebraically

Use the procedure for attaining an inverse algebraically on one of the two

functions to distinguish if they are inverses.

Example:

Determine if the functions y = 3x2 – 12x + 15, x ≥ 2 and

are inverses

P.52 – 55 #3 – #6, #10, #12, #14 – #16, #20


Also

State the restricted domain for each of the following relations and so that the

inverse relation is a function, and write the equation of the inverse.

(i) y = x2 – 6x + 10 (ii) y = 5x

2 + 20x – 9

(iii) y = 2x2 – 8x + 1 (iv) f(x) = 3x

2

(v) f(x) = x2 + 2x (vi) f(x) = 2x

2 + 4

1.1 horizontal & vertical translations - · pdf fileunit ii transformations of functions 1...

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