4.3 Reflecting Graphs; Symmetry Notes
Precalculus
Reflection in the x-axis.
The graph of y = -f(x) is obtained by reflecting the graph of y = f(x) across the xaxis. For the two graphs below, construct y = -f(x).
(For reflection across the x-axis, point (x, y) becomes (x, -y)).
8
y
y = x2 – 3
6
6
4
y = f(x)
4
2
-6
-4
2
-2
4
2
6 x
-2
-4
-6
2
For graphing the absolute value of a function, the graph stays the same when y
is positive. If y is negative, x stays the same but y becomes positive.
Graph y = f  x 
8
y
6
6
4
4
y = f(x)
2
2
-6
-4
2
-2
4
6 x
-2
-4
2
-6
y = x2 – 3
Reflection in the y-axis.
The graph of y = f(-x) is obtained by reflecting the graph of y = f(x) across the yaxis. Each point (x, y) on the original graph becomes the point (-x, y) on the
reflected graph.
8
g(x) = 1.5x
8
6
6
4
4
2
2
5
5
h( x ) = ( x + 3 )
2
Reflection in the line y = x
Reflecting the graph of an equation in the line y = x is the same as interchanging
x and y in the equation. For reflection across the line y = x, each point (x, y)
becomes the point (y, x).
8
8
g(x ) = x 2
3∙x
g( x ) = x 2
6
6
4
4
2
2
5
3∙x
5
2
2
y=x
Symmetry
A line l is called an axis of symmetry if it is possible to pair the points of the
graph in such a way that l is the perpendicular bisector of the segment joining
each pair.
A point O is called a point of symmetry of a graph if it is possible to pair the
points of the graph in such a way that O is the midpoint of the segment joining
each pair.
l
O
Point of symmetry
Axis of symmetry
Symmetry in the origin
(-x, -y) is on the graph whenever (x, y) is on the graph. y = x3 is an example of
symmetry in the origin.
(x, y)
(-x, -y)
Section 4.3 Reflected Graphs
Precalculus
The graph of y = f(x) is given. Sketch the graphs of: y = -f(x), y  f  x  ,
and y = f(-x).
1.)
a.) y = -f(x)
y
y
2
-4
2
2
-2
4
6 x
-4
2
-2
-2
c.) y = f(-x)
y
y
2
2
2
-2
4
6 x
-4
6 x
a.) y = -f(x)
y
2
2
2
-2
4
6 x
-4
6 x
c.) y = f(-x)
y
y
2
2
2
-2
4
-2
b.) y  f  x 
-2
2
-2
-2
-4
4
-2
y
-4
2
-2
-2
2.)
6 x
-2
b.) y  f  x 
-4
4
4
6 x
-4
2
-2
-2
4
6 x
3.)
a.) y = -f(x)
y
y
2
2
-4
2
-2
4
-4
6 x
b.) y  f  x 
6 x
c.) y = f(-x)
y
y
2
2
2
-2
4
-2
-2
-4
2
-2
4
6 x
-4
2
-2
-2
4
6 x
-2
4.) Sketch the graphs of y = x2 – 9, y = 9 – x2, and y = 9  x 2 on a single set of
axes. (Note that if f(x) = x2 – 9, this is the same as y = f(x), y = -f(x), and
y
y   f  x  ).
10
8
6
4
2
-8
-6
-4
2
-2
-2
-4
-6
-8
-10
4
6
8 x
5.) Sketch the graphs of y = x  2 , y = 2  x , and y = 2  x on a single set of
axes. (Note that if f(x) = x  2 , this is the same as y = f(x), y = -f(x), and
y
10
y   f  x  ).
8
6
4
2
-8
-6
-4
2
-2
4
6
8 x
-2
-4
-6
-8
-10
Sketch the graph of each equation and the reflection of the graph in the line y =
x. then give an equation of the reflected graph.
y
6.) y = 3x – 4
4
2
Equation of reflected graph
-6
-4
-2
2
-2
__________________________
-4
4
6 x
7.) y = x2 – 2x
y
4
2
-6
-4
-2
2
4
6 x
2
4
-2
-4
Equation of reflected graph
_________________________
8.) y = x  2
y
4
2
-6
-4
-2
-2
Equation of reflected graph
_________________________
-4
6 x
A function f(x) is an even function if f(-x) = f(x)
A function f(x) is an odd function if f(-x) = -f(x)
10.) classify each function as even, odd, or neither
a.) f(x) = x2 ____________
b.) f(x) = x3 _____________
c.) f(x) = x2 – x ___________
d.) f(x) = x4 + 2x2 ____________
e.) f(x) = x3 + 3x2 ___________
f.) f(x) = x5 – 4x3 ____________
11.) What kind of symmetry does the graph of an even function have?
_____________________________
12.) What kind of symmetry does the graph of an odd function have?
_____________________________
If there is symmetry:
in the x-axis:
in the y-axis:
in the line y = x:
in the origin (0,0):
(x, -y) is on the graph whenever (x, y) is
(-x, y) is on the graph whenever (x, y) is
(y, x) is on the graph whenever (x, y) is
(-x, -y) is on the graph whenever (x, y) is
13.) Use symmetry to graph the following:
a.) x  y  2
b.) x2y = 1
y
-6
-4
y
4
4
2
2
-2
2
4
6 x
-6
-4
-2
2
-2
-2
-4
-4
4
6 x