I. Symmetry
In mathematics, a knowledge of symmetry is helpful when
graphing and analyzing equations and
functions .
A. x-axis symmetry
The graph of an equation is said to be symmetric with
respect to the x-axis if, for any point (x, y) on the graph,
there is matching point (x, –y).
To test an equation for x–axis symmetry, we replace
y with –y, simplify, and then compare the resulting equation
with the original equation to see if they are equivalent.
The graph of an equation with x-axis symmetry looks
like it can be folded along the x-axis and the parts of the
graph above and below the x-axis will coincide.
Equations with x-axis symmetry are not functions. |
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B. y-axis symmetry
The graph of an equation is said to be symmetric with
respect to the y-axis if, for any point (x, y) on the graph,
there is matching point (–x, y).
To test an equation for y–axis symmetry, we replace
x with –x, simplify, and then compare the resulting equation
with the original equation to see if they are equivalent.
The graph of an equation with y-axis symmetry looks
like it can be folded along the y-axis and the parts of the
graph to the left and right of the y-axis will coincide. |
 |
C. origin symmetry
The graph of an equation is said to be symmetric with
respect to the origin if, for any point (x, y) on the graph,
there is matching point (–x, –y).
To test an equation for origin symmetry, we replace
x with –x AND y with –y, simplify, and then compare the
resulting equation with the original equation to see
if they are equivalent.
The graph of an equation with origin symmetry looks
like it can be rotated 180° around the origin and the
parts of the graph will coincide. |
 |
Note: When you replace x with –x and/or y with – y
and then simplify , if it looks like all the signs in the equation
changed or n one of the signs in the equation changed, then the resulting
equation is equivalent to the
original . If it looks like some signs changed and others did not, the resulting
equation is not equivalent to
the original equation.
Example 1: Algebraically determine whether the
graph of the equation xy – x^2 =3 is symmetric with
respect to the x-axis, the y-axis, and/or the origin. (#26 p. 163)
Test for x-axis symmetry: Replace y with –y. Simplify.
x(–y) – x^2 = 3 → –xy – x^2 = 3 Only first sign changed →
not symmetric to x-axis
Test for y-axis symmetry: Replace x with –x. Simplify
(–x)y – (–x)^2 = 3 → –xy – x^2 = 3 Only first sign changed
→ not symmetric to y-axis
Test for origin symmetry: Replace x with –x AND y with –y.
Simplify
(–x)(–y) – (–x)^2 = 3 → xy – x^2 = 3 No signs changed →
symmetric with respect to origin
II. Even and Odd Functions
If the graph of a function f is symmetric with respect to
the y-axis, we say that it is an even function.
That is, for each x in the domain of f, f(–x) = f(x).
If the graph of a function f is symmetric with respect to
the origin, we say that it is an odd function.
That is, for every x in the domain of f, f(–x) = – f(x).
To determine of a function is odd or even, find f(–x),
i.e. replace x with –x, and simplify.
If the resulting ex pression on the right looks like no
signs have changed, the function is even.
If the resulting expression on the right looks like all signs have changed,
the function is odd.
If the resulting expression on the right looks like some signs have changed,
the function is neither even nor odd.
Example 2: Algebraically determine whether the
function
is even, odd, or neither. (#48 p.
164)
no signs changed → f
is an even function → f has y-axis symmetry
III. Review of Basic Functions
IV. Transformations
A. Vertical Shifts
To determine whether there has been a vertical shift, look for a constant, k,
being added to or
subtracted from the base function (on the outside). If k is positive the shift
is up; if k is negative
the shift is down.
The graph of y = f(x) + k is the graph of f(x) shifted up k units.
The graph of y = f(x) – k is the graph of f(x) shifted down k
units.
B. Horizontal Shifts 
To determine whether there has been a horizontal shift,
look for a constant, h, being added to or
subtracted from the x itself (on the inside).
The graph of y = f(x – h) is the graph of f(x) shifted right h
units. h is positive.
The graph of y = f(x + h) is the graph of f(x) shifted left h
units. h is negative.
To find the value of h, set the expression in the parentheses equal to zero and
solve for x .
If h is positive the shift is to the right; if h is negative, the shift is to
the left
C. Key Point (h, k)
For transformed functions of the form g(x) = f(x – h) + k,
(h, k) is a key point on the graph of f.
For quadratic, quartic, and absolute value functions, (h, k) is the vertex.
For cubic and cube root functions, (h, k) is the inflection point.
To graph these types of functions, we make (h, k) the center of our t-chart.
We then choose two x-values to the left of h and two x-values to the right of h.
For square root functions, (h, k) is the point where the graph either begins or
ends. To graph
these types of functions, we make (h, k) either the first or the last point in
our t-chart.
Example 3:
Describe how
the graph of g(x) = (x + 1)^2 – 1 can be obtained from
one of the basic functions. Then graph the function. (#63 p. 164) |
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The graph of g(x) can be
obtained by shifting
the graph of y = x^2 one unit left and one
unit down. The result will be a parabola
opening up with its vertex at (–1, –1). |
 |
D. Reflections y = –f(x) or y = f(–x)
To determine whether or not there has been a reflection,
we look for a negative coefficient.
If the negative is on the outside, the function has an x-axis reflection. If the
negative is on the
inside (i.e., it is touching x), the function has a y-axis reflection.
The graph of y = –f(x) is the graph of f(x)
reflected across the x-axis
Hint: Functions which have been reflected across the
x-axis open down or go downhill from left to right.
The graph of y = f(–x) is the graph of f(x) reflected across the
y-axis.
Hint: Square root functions which have been reflected
across the y-axis open left. Cubic and cube
root functions which have been reflected across the y-axis go downhill from left
to right.
Example 4:
Describe how the graph of
can be obtained from one of the basic
functions
Hint: Before attempting to identify the transformations,
it might be helpful to rewrite the given
function as
f(x) can be obtained by reflecting the graph of y = 1/x
over the x-axis and shifting it up 5.
E. Vertical Stretching and Shrinking y = a·f(x)
If |a| > 1, the graph of y = a·f(x) is the graph of f(x)
vertically stretched by a factor of a.
If 0 < |a| < 1, the graph of y = a·f(x) is the graph of f(x) vertically
shrunk by a factor of a.
Hint: To determine whether there has been a vertical
stretch or shrink, ignore the sign on a. Just look
at the number. If |a| is larger than 1, there has been a vertical stretch. If
|a| is between 0 and 1,
there has been a vertical shrink.
If a is negative, the graph of f(x) has also been
reflected across the x-axis.
F. Horizontal Stretching and Shrinking y = f(c·x)
If |c| > 1, the graph of y = f(c·x) is the graph of
f(x) horizontally shrunk by a factor of 1/c.
If 0 < |c| < 1, the graph of y = f(c·x) is the graph of f(x)
horizontally stretched by a factor of 1/c
Hint: To determine whether there has been a horizontal
stretch or shrink, ignore the sign on c. Just
look at the number. If |c| is larger than 1, there has been a horizontal shrink.
If |c| is between
0 and 1, there has been a horizontal stretch.
If c is negative, the graph of f(x) has been also
reflected across the y-axis.
Example 5:
Describe how the graph of
can be obtained from one of the basic
functions.
Then graph the function.
It might be helpful to rewrite the given function as
before attempting to identify
the transformations.
g(x) can be obtained by shifting the graph of
four to the left, reflecting the graph
over the y-axis, and horizontally shrinking it by 1/2
The graph of g(x) is a half parabola opening left with
its end-point at (2, 0).
Hint: To find h, set –2x + 4 = 0 and solve for x.
V. How to Write the Equation of a Transformed Function
A. From its description
Example 6:
Write the equation for a function with the following
characteristics
The shape of y = x^4, but upside down, stretched
vertically by a factor of 3, and shifted
right five and up one. (like #97 – 106 p. 164)
“upside down” implies there has been an x-axis reflection
→ y = –x^4
“stretched vertically by a factor of 3” → y = –3x^4
“shifted right 5” → y = –3(x – 5)^4
“shifted up 1” → y = –3(x – 5)^4 + 1
B. From its graph
Example 7:
Find a formula for the following transformation of the
graph of f(x) = x^3 – 3x^2. (#132 p. 166)
| original function: f(x) |
transformed function: g(x) |
 |
 |
Comparing the points on the transformed graph g(x) to the
points on the original function
f(x), we see that the x-values have increased by 2 and the y-values have
increased by 1.
This means that the graph of f(x) has been shifted two units right and one unit
up.
Therefore, g(x) = f(x – 2) + 1 = (x – 2)^3 – 3(x – 2)^2 + 1.
