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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
Unit II Transformations of Functions 2
Function Table of Values Graph
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
_________________
Unit II Transformations of Functions 3
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.
Unit II Transformations of Functions 4
(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.
Function Table of Values Graph
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
Unit II Transformations of Functions 5
Function Table of Values Graph
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)
Unit II Transformations of Functions 6
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
Unit II Transformations of Functions 7
(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)
Unit II Transformations of Functions 8
(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) ( , )
Unit II Transformations of Functions 9
(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
Unit II Transformations of Functions 10
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
Unit II Transformations of Functions 11
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.
Unit II Transformations of Functions 12
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
Unit II Transformations of Functions 13
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:
Unit II Transformations of Functions 14
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
Unit II Transformations of Functions 15
(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
Unit II Transformations of Functions 16
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)
Unit II Transformations of Functions 17
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)
Unit II Transformations of Functions 18
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
Unit II Transformations of Functions 19
Example:
Below is the graph of y = f(
x) [or y = sin (
x)].
Produce the table of values and state the mapping rule.
Mapping Rule: (x, y) ( )
Example:
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 ° 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
Unit II Transformations of Functions 20
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
Unit II Transformations of Functions 21
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
Unit II Transformations of Functions 22
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).
Unit II Transformations of Functions 23
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)
Unit II Transformations of Functions 24
Example:
Given the graph of y = f(x), sketch the graph of y + 2 = 2f(–3x – 3) on the same
grid.
Unit II Transformations of Functions 25
(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) ______
Unit II Transformations of Functions 26
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 ________________________________
Unit II Transformations of Functions 27
Example:
Determine the specific equation for the image of y = f(x) in the
form y = af(b(x – h)) + k.
Unit II Transformations of Functions 29
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
Unit II Transformations of Functions 30
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
Unit II Transformations of Functions 31
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)
Unit II Transformations of Functions 32
(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) → (
)
Unit II Transformations of Functions 33
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 ____________)
Unit II Transformations of Functions 34
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.
Unit II Transformations of Functions 35
(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?
Unit II Transformations of Functions 36
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?
Unit II Transformations of Functions 37
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
Unit II Transformations of Functions 38
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
Unit II Transformations of Functions 39
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)
Unit II Transformations of Functions 40
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
Unit II Transformations of Functions 41
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