Math 30-1
Transformations
Specific Outcome: Students will demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations.
Achievement Indicators:
Compare the graphs of a set of functions of the form to the graph of , and generalize, using inductive reasoning, a rule about the effect of k. Compare the graphs of a set of functions of the form to the graph of , and generalize, using inductive reasoning, a rule about the effect of h. Compare the graphs of a set of functions of the form to the graph of
, and generalize, using inductive reasoning, a rule about the effect of h and k. Sketch the graph of , , or for given values of h
and k, given a sketch of the function , where the equation of is not given.
Write the equations of a function whose graph is a vertical and/or horizontal translation of the graph of the function .
Specific Outcome: Students will demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations.
Achievement Indicators:
Compare the graphs of a set of functions of the form to the graph of , and generalize, using inductive reasoning, a rule about the effect of a.
Compare the graphs of a set of functions of the form to the graph of , and generalize, using inductive reasoning, a rule about the effect of b.
Compare the graphs of a set of functions of the form to the graph of , and generalize, using inductive reasoning, a rule about the effect of a and b.
Sketch the graph of , , or for given values of a and b, given a sketch of the function , where the equation of is not given.
Write the equations of a function, given its graph which is a vertical and/or horizontal stretch of the graph of the function .
Specific Outcome: Students will apply translations and stretches to the graphs and equations of functions.
Sketch the graph of the function for given values of a, b, h, and k, given the graph of the function , where the equation of is not given.
Write the equation of a function, given its graph which is a translation and/or stretch of the graph of the function .
Specific Outcome: Students will 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, and line
Achievement Indicators:
Generalize the relationship between the coordinates of an ordered pair and the
coordinates of the corresponding ordered pair that results from a reflection through the x -axis, y-axis or the line .
Sketch the reflection of the graph of a function through the x-axis, the y-axis or the line , given the graph of the function , where the equation of
is not given.
Generalize, using inductive reasoning, and explain rules for the reflection of a graph of the function through the x-axis, y-axis or the line .
Sketch the graphs of the functions , and , given the graph of the function , where the equation of is not given.
Write the equation of a function, given its graph which is reflection of the graph of the function through the x-axis, y-axis or the line .
Specific Outcome: Students will demonstrate an understanding of inverses of relations. Achievement Indicators:
Explain how the graph of the line can be used to sketch the inverse of a relation.
Explain how the transformation can be used to sketch the inverse of a relation.
Sketch the graph of the inverse relation, given the graph of a relation.
Determine if a relation and its inverse are functions.
Determine restrictions on the domain of a function in order for its inverse to be a function.
Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation.
Explain the relationship between the domains and ranges of a relation and its invers.
Determine, algebraically or graphically, if two function are inverses of each other.
Specific Outcome: Students will graph and analyze radical functions (limited to functions involving one radical).
Lesson 1: Horizontal and Vertical Translations
Interval Notation
There are 2 equivalent ways of indicating number line intervals:
There are 6 “Basic Functions” that we will be transforming in this unit.
Problem 1: With the assistance of a graphing calculator, sketch the graphs of the functions below. Then analyze these functions by stating the domain, range, intercepts, and asymptotes.
a) Quadratic: b) Cubic:
c) Square Root:
Set Notation Interval Notation
x axb
a,b x axb
[a,b]x xa
[a,) x xb
,b
,b) Cubic:
Domain: ____________ Domain: __________ Domain: ____________
Range: _____________ Range: ___________ Range: _____________
x-int: ______________ x-int: ____________ x-int: ______________
y-int: ______________ y-int: ____________ y-int: ______________
asymptotes: _________ asymptotes: _______ asymptotes: _________
d) Reciprocal: e) Exponential: f) Absolute Value
Domain: ____________ Domain: __________ Domain: ____________
Range: _____________ Range: ___________ Range: _____________
x-int: ______________ x-int: ____________ x-int: ______________
y-int: ______________ y-int: ____________ y-int: ______________
asymptotes: _________ asymptotes: _______ asymptotes: _________
Define: A translation is an operation which causes the graph of a function to “shift” horizontally or vertically.
In General: In the equation y = f(x), replacing y with y – k results in a translation…
k units ______________ if k > 0, and
k units ______________ if k < 0.
This is sometimes shown in mapping notation: (x,y)(x,yk)
Problem 3: Describe how the graph of is related to the graph of .
Problem 4: Consider the function whose graph is shown below. Write the equation, and sketch the graph if we make the following changes to the equation:
In General: In the equation y = f(x), replacing x with x – h results in a translation…
h units ______________ if h > 0, and
h units ______________ if h < 0.
This is sometimes shown in mapping notation: (x,y)(xh,y)
Problem 6: Consider the function . Describe how
the graph of will be related to the graph of , and use this information to sketch the graph of
on the grid provided.
Problem 7: The function y=
1
x+2 undergoes a vertical
translation of 3 units down and a horizontal translation of 5 units to the right. Write the equation of the transformed function.
Problem 8: What vertical translation is applied to
y
=
√
x
so that the transformed function passes through the point(
25
,
−
1
)
.Problem 9: Describe the transformations on y f(x) indicated by the mapping notation
Assignment: Page 169 #4-15 (every second one), MC #1,2
Lesson 2: Vertical and Horizontal Stretches
Define: A stretch is an operation which causes the graph of a function to expand or compress horizontally or vertically.
Investigate: The graph of the function is shown below. Write the equation, and sketch the graph, given the following changes to the equation:
State the coordinates of any invariant points.
Function Notation Function Notation
In General: In the equation y f(x) the transformation results in a stretch by a factor of ________ about the __________________.
Investigate: The graph of the function is shown below. Write the equation, and sketch the graph given the following changes to the equation:
State the coordinates of any invariant points.
In General: In the equation y f(x)the transformation results in a stretch by a factor of ________ about the __________________.
Problem 2: Write the replacement for x and/or y, then write the equation of the image of
if the graph is:
a) stretched horizontally about the y-axis by a factor of 2.
b) Stretched horizontally about the y-axis by a factor of and vertically about the x-axis by a
factor of .
Problem 3: Describe how the graph of the second function will compare to the graph of the first function:
a) y x
1
3y x b) y2 x y6 x c) y x 3 y 3x3 Function Notation
Mapping Notation Function Notation
Assignment: Page 201 #3a,4,5a,6,8a,9b,10,14,15 and Page 211 MC#2
Lesson 3: Reflecting Graphs of Functions
Define: A reflection is an operation which causes the graph of a function to “flip” across a line of axis.
Vertical Reflections
Problem 1: The partial graph of is shown below.
a) Write the equation, and sketch the graph if we make the following change to the equation:
b) State the coordinates of any invariant points.
In General: In the equation , the replacement results in a reflection _______________________________________.
Problem 2: The graph of is shown on the right. Sketch the graph of and state the coordinates of any invariant points.
Horizontal Reflections
Problem 3: The partial graph of is shown below.
a) Write the equation, and sketch the graph if we make the following change to the equation:
b) State the coordinates of any invariant points.
In General: In the equation the replacement results in a reflection __________________________________________.
Problem 4: The graph of is shown on the right. Sketch the graph of and state the coordinates of any invariant
points.
Problem 5: Describe how the graph of the second function compares to the graph of the first function:
a) b) y
1
x1y
1
x1
Assignment: Page 183 #3-6,7,8,10,MC#1,2 Page 202 #3b,5bc,7,8b,9a,11,MC #1
Lesson 4: Combined Transformations
Recall: Write the equation of the “new” function, and describe how its graph will be transformed if the following changes are made to the equation of :
In General: The graph of will be related to the graph of as
follows:
Statethe order that these transformations should be performed:
Problem 1: The graph of has been stretched horizontally by a factor of about the y-axis and translated 5 units left. State the equation for the transformed function.
Problem 3: Sketch the graph of the new function above and write the equation of the transformed function
Problem 4: The key point (2,-3) lies on the graph of . Find the coordinates for the image of this point given the following:
Problem 5: Consider the function . Describe how the graph of will be transformed to yield this new function.
Problem 6: For each of the following, the thick graph is a transformation of the thin graph. Describe the transformation. Give the equation of the thick graph. List any invariant points.
a) b)
Transformation: Transformation:
Equation: Equation:
x y
x y
y f(x)
Problem 7: The graph of the function y = g(x) represents a transformation of the graph of y = f(x). Determine the equation of g(x) in the form
.
Problem 8: The graph of the function is shown on the right.
a) What are the zeros of the function?
b) Use transformations to determine the zeros of the following functions:
Lesson 5: Inverse Relations
Define: The inverse of a relation is formed by switching the x-coordinates with the y -coordinates for every point in the relation. The inverse of is symbolized by
. If the inverse of is also a function then it can be symbolized by
.
Problem 1: The partial graph of is shown on the
right.
a) Write the equation, and sketch the graph which represents .
b) State the coordinates of any invariant points.
c) State the domain and range of and .
d) How could the domain be restricted on so that the inverse is also function?
e) Sketch the “mirror line” for this transformation.
In General:
In the equation interchanging x and y results in a reflection ___________ _______________________.
The domain of becomes the __________ of and vice versa.
The ________________________ test can be used to determine if an inverse will be a function.
Problem 3: The graph of is shown on the right:
a) Sketch the inverse of
b) Is the inverse a function?
c) Are there any invariant point(s)?
d) Compare the domain and range of with its inverse.
e) Sketch the line
