A reflection is like
a mirror image. The line of
reflection
acts as the mirror and is halfway between the point and its
image.
If the point is its own reflection, then it is a point on the line of
reflection.
Some of the important relections are listed below with examples.

1) y = -f(x) (This
is the reflection about the x-axis of the graph y = f(x).) That
is
for every point (x, y) there is a point (x, -y).

Look at the example
graphs below of y = x^{2} and y = -x^{2}.

Notice the green
graph is simply the same as the blue
graph folded down across the x-axis.
The x-axis is the line of reflection or the mirror.

2) y = |f(x)| means
that the entire graph will be above the x-axis. Why? (The
absolute value is always positive, that's why!!)
To graph the absolute value graph, graph the function y = f(x).
Anything
above the x-axis, stays above it, anything below the x-axis is
reflected
above the x-axis and anything on the x-axis, stays on the x-axis.
Study the following graphs that illustrate:

....................

y = x^{2} -
4
y = |x^{2} - 4|

The absolute value of
the left graph is found by taking all points above the x-axis and
leaving
them above the x-axis. These are the purple points. The
points
below the x-axis are reflected up above the x-axis. These are the
red points. Of course, the points on the x-axis stay on the
x-axis.

3) y = f(-x)
(This is reflection about the y-axis of the graph y = f(x)) For
every
point on the right of the y-axis, there is a point equidistant to the
left
of the y-axis. That is for every point (x, y), there is a point
(-x,
y). Study the graphs below:

Notice that every point on the left of the y-axis has an image on the
right
of the y-axis. Look at the equations. The only thing that
makes
them different is the negative sign in front of only the x
variable.
That is what makes them reflections about the y-axis.

4) Reflections
about the line y = x is accomplished by interchanging the x and the
y-values.
Thus for y = f(x) the reflection about the line y = x is accomplished
by
x = f(y). Thus the reflection about the line y = x for y = x^{2}
is the equation x = y^{2}. These are graphed on the
following
graph:

The blue line is where
you would fold the paper for these two graphs to match. Notice
that
the green parabola is opening up but when you fold it across the blue
line,
it opens to the right.

Symmetry

A line is called an axis
of symmetry of a graph if you can pair the
points such that the line is perpendicular of the segment joining the
pair
of points. Study the graph below to identify the axis of symmetry:

Ways to test different
symmetry

Types of symmetry

Symmetry to the x-axis
(x, -y) is on the graph when (x, y) is.

Test for symmetry
In the equation, leave x alone and replace
y with -y. If you get the same equation, it is symmetric to the
x-axis.

Symmetry to the y-axis
(-x, y) is on the graph when (x, y) is.

Test for symmetry.
In the equation, replace x with
-x. If the equation is the same, it is symmetric to tye y-axis.

Symmetry to the line y = x.
(y, x) is on the graph when (x, y) is.

Test for symmetry.
Interchange x and y. If the equation
is the same then it is symmetric to the line y = x.

Symmetry to the origin.
(-x, -y) is on the graph when (x, y) is.

Test for symmetry.
Replace x with -x and y with -y. If
the equation is the same, then it is symmetric to the origin.

Examples

1) Find the symmetry
for the equation x^{2} + y^{2} = 9.

Symmetry to x-axis

Symmetry to y-axis

Symmetry to y = x

Symmetry to origin

x^{2} + (-y)^{2} = 9
x^{2} + y^{2} = 9

(-x)^{2} + y^{2} = 9
x^{2} + y^{2} = 9

y^{2} + x^{2} = 9
x^{2} + y^{2} = 9

(-x)^{2} + (-y)^{2} = 9
x^{2} + y^{2 }= 9

yes

yes

yes

yes

2) Find the symmetry
for the equation x^{4} = y - 3.

Symmetry to x-axis

Symmetry to y-axis

Symmetry to y = x

Symmetry to origin

x^{4 }= (-y) - 3
x^{4} = -y - 3

(-x)^{4} = y - 3
x^{4} = y - 3

y^{4} = x - 3

(-x)^{4} = (-y) - 3
x^{4 }= -y - 3

no

yes

no

no

Graphing Problems

1) Graph
(blue)
Find and graph the equation of :

a) Reflection about x-axis (green)
b) Reflection about y-axis (purple)
c) Reflection about y = x (red)

Solutions:

2) Use symmetry
to graph |x| + |y| = 1

Symmetry to x-axis

Symmetry to y-axis

Symmetry to y = x

Symmetry to origin.

|x| + |-y| = 1
|x| + |y| = 1

|-x| + |y| = 1
|x| + |y| = 1

|y| + |x| = 1
|x| + |y| = 1

|-x| + |-y| = 1
|x| + |y| = 1

yes

yes

yes

yes

Because we have symmetry
to all four, we have a reflection in quadrant II, III and IV.
Thus,
all you need to do is make the graph in Quad I and reflect into the
other
three quadrants. The equation in quad I is x + y = 1

Now reflect into
the other three quadrants:

Sail
towards the next section:

Sail
back to the previous section:

Current quizaroo
# 4a

1) Give the domain for the function:
f(x)
= (x - 3)/(x - 1)

a)
All real numbers
but 3 and 1

b)
All real numbers

c)
All real numbers
but 3

d)
All real numbers
but 1

e)
All real numbers
between 1 and 3

2) Give the range for the
function: f(x) = | x |
+ 1

a)
y > 0

b)
y >
1

c)
All real numbers

d) x
>
0

e) x
>
1

3) Given f(x) = x^{2},
g(x) = 2x and h(x) = x + 3, find f(g(h(2))) =

a)
100

b) 11

c) 49

d) 50

e) 19

4) Tell the
symmetry for y^{2} + xy = 10. Use
a) x-axis, b) y-axis, c) line y = x, d)
origin