Section 4-3:  Reflections and Symmetry 

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Reflections
 
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 = x2 and y = -x2.
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 = x2 - 4                                            y = |x2 - 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 = x2 is the equation x = y2.  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 x2 + y2 = 9.
 
Symmetry to x-axis Symmetry to y-axis Symmetry to y = x Symmetry to origin
x2 + (-y)2 = 9 
x2 + y2 = 9
(-x)2 + y2 = 9 
x2 + y2 = 9
y2 + x2 = 9 
x2 + y2 = 9
(-x)2 + (-y)2 = 9 
x2 + y2 = 9
yes yes yes yes
2)  Find the symmetry for the equation x4 = y - 3.
 
Symmetry to x-axis Symmetry to y-axis Symmetry to y = x Symmetry to origin
x4 = (-y) - 3 
x4 = -y - 3
(-x)4 = y - 3 
x4 = y - 3
y4 = x - 3 (-x)4 = (-y) - 3 
x4 = -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) = x2, 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 y2 + xy = 10.  Use a)  x-axis, b) y-axis,  c)  line y = x,  d)  origin
a)  a only
b)  b only
c)  c only
d)  d only
e)  not symmetric to the above four
 
 
  5)  Classify the function f(x) = | x | 
 
a)  odd function
b)  even function
c)  both even and odd
d)  neither even or odd
e)  symmetric to origin
 
 
 
 
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