Download Pure Math 30 Transformations Review

Math 30-1 Ch. 1 Review
Review
1. Linear Functions:
 y = mx + b
 1st degree equation
 straight line
 Domain: x ε R, Range: y ε R
2. Quadratic Functions:
 y  x2
 second degree equation
 shaped as a parabola
 Domain: x ε R, Range: y ≥ 0
3. Cubic Functions:
 y  x3
 third degree equation
 Domain: x ε R, Range: y ε R
4. Absolute Value Functions:
 y x
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shaped as a V
Domain: x ε R, Range: y ≥ 0
5. Rational Functions:
1
 y
x
 restrictions or non-permissible values/asymptotes
 Domain: x ≠ 0, x ε R, Range: y ≠ 0, y ε R
6. Radical Functions:
 y x
 The radicand must be positive
 Domain: x ≥ 0, Range: y ≥ 0
Horizontal and Vertical Translations
a. Vertical Translation:
y = f(x) + k
OR
y – k = f(x)
If k > 0, the function moves up
If k < 0, the function moves down.
Mapping: (x, y) goes to (x, y + k)
b. Horizontal Translation: y = f(x – h)
For x – h, the function moves right
For x + h, the function moves left
Mapping: (x, y) goes to (x + h, y)
Reflections
a.
y = -f(x)
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b.
all y-values change their sign
the x-axis is the mirror reflection line (flipped over the x-axis)
the zeros don’t change (invariant points)
In general, (x, y) becomes (x, -y).
y = f(-x)
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all x-values change their sign
the y-axis is the mirror line (flipped over the y-axis)
the y-intercept is invariant
In general, (x, y) becomes (-x, y)
Stretches
a.



vertical: y=af(x)
Multiply each y value by a.
In general, (x, y) becomes (x, ay)
If a < 0, the graph is also reflected in the x-axis.
b. horizontal: y = f(bx)
 Divide the x-value at each point on the graph (i.e. multiply by reciprocal)
x
 In general, (x, y) becomes ( , y )
b
 If b < 0, the graph is also reflected in the y-axis.
 Remember that the stretch is always stated as the reciprocal. I.e. f(2x)
is stated as horizontal stretch by factor of 1/2
Combinations of Transformations
Recap:
a:
b:
h:
k:
y = af(b(x – h)) + k
vertical stretch/reflection in the x-axis
horizontal stretch/reflection in y-axis
horizontal transformation
vertical transformation
Be able to do these transformations both on your calculator and by hand. And
remember to read them from left to right.
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
Unless otherwise indicated, use the following order to describe how to transform
from one graph to another.
1.
Stretches
2.
Reflections
3.
Translations
Or in other words, if you are looking at the equation, go from left to right. (since
order doesn’t matter when reflections and stretches are combined)
Inverse Functions
y f

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1
( x) OR x = f(y)
this is an inverse function
switch the x and y and solve for y
the mirror line (reflecting line) is y = x
In general, (x, y) becomes (y, x)
With an inverse, if the original function has a domain A and range B, the inverse
function will have a domain B and a range A.
To test if something is a function, use the vertical line test. If the vertical line hits
the graph at only one point, it is a function.
To find the equation of an inverse algebraically, follow two steps:
1. Switch the x and y in the equation.
2. Solve for y.
To restrict the domain so that an inverse is a function, look at the original domain
and how it compares to the inverse. In a parabola, you would only take half of
the original parabola so that when you draw the inverse, you get a function.
Things Alberta Education wants you to know about Relations and
Functions:
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Transformations include the 6 base functions as well as exponents and
logarithms
Analyzing a transformation for includes: determining and describing the
effects of a transformation on the domain, range, intercepts, invariant
points, and key points.
The term “key points” on a relation or function may include vertices,
starting points, maximum and minimum points, etc.
Mapping notation may be used for transformations.
Set builder notation and interval notation are acceptable ways to express
domain and range.
Stretches and reflections are performed prior to translations unless
otherwise stated.
Students should understand the concept of invariant points.
Reflections in the line y = x should not be combined with any other
transformations.
Students should be familiar with the notation x = f(y)



1
The notation y  f ( x) should only be used if the inverse is also a
function.
When discussing the equations of inverse relations, the focus should
primarily be on linear or quadratic functions.
Students should be able to restrict the domain on the original function to
obtain an inverse that is also a function.