polynomial functions
MHF4U: Advanced Functions
Transformations of Polynomial Functions
J. Garvin
Slide 1/18
polynomial functions
Transformations
In most cases, the graph of a function is similar to a simpler
version, but may appear stretched, shifted or reflected to
some extent.
The simplest version of a function that possesses all of the
same characteristics of the derived function is called a parent
function or a base function.
If we know information about a particular base function, it
may be possible to sketch a graph of the derived function by
analyzing the transformations that have been applied to the
base function.
J. Garvin — Transformationsof Polynomial Functions
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polynomial functions
Transformations
Transformations of Polynomial Functions
A polynomial of the form f(x) = a(b(x−c))n+d, where a,
b,cand dare real constants, and nis a natural number, is a
transformation of some power function g(x) = xn.
In the form above:
•ais a vertical stretch/compression, and possibly a
reflection
•bis a horizontal stretch/compression, and possibly a
reflection
•cis a horizontal translation
•dis a vertical translation
J. Garvin — Transformationsof Polynomial Functions
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polynomial functions
Vertical and Horizontal Transformations
Transformations may be applied vertically or horizontally.
In the function f(x) = a(b(x−c))n+d, both aand dare
vertical transformations. They appear “outside” of the
function in its equation.
The parameters band care horizontal transformations. They
appear “inside” of the function’s equation.
Horizontal transformations may appear to behave opposite to
intuition: larger numbers for bcompress the graph, smaller
numbers stretch it, and the parameter cseems to shift the
graph in the opposite direction.
J. Garvin — Transformationsof Polynomial Functions
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polynomial functions
Vertical Stretches/Compressions
Example
Sketch graphs of f(x)=2x3and g(x) = 1
3x3.
For f(x), |a|>1, so it has a vertical stretch by a factor of 2.
All points are twice as far from the x-axis as they are on the
graph of y=x3.
For g(x), 0 <|a|<1, so it has a vertical compression by a
factor of 1
3. All p oints are one-third as far from the x-axis as
they are on the graph of y=x3.
J. Garvin — Transformationsof Polynomial Functions
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polynomial functions
Vertical Stretches/Compressions
J. Garvin — Transformationsof Polynomial Functions
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