Lesson 3-1

Example 1
Determine whether each graph is symmetric with respect to the origin.

a.	f(x) = 	b.	g(x) = 	
				
	The graph of f(x) =  appears to be symmetric with respect to the origin.		The graph of g(x) =  does not appear to be symmetric with respect to the origin.	

We can verify these conjectures algebraically by following these two steps.

1.	Find f(-x) and - f(x).
2.	If f(-x) = - f(x), the graph has point symmetry.

a.	f(x) = 
	Find f(-x).
			Find -f(x).

The graph of f(x) =  is symmetric with respect to the origin because f(-x) = - f(x).

b.	g(x) = 
	Find g(-x).
			Find -g(x).

The graph of g(x) =  is not symmetric with respect to the origin because g(-x) ≠ -g(x).

Example 2
Determine whether the graph of  = -3 is symmetric with respect to the x-axis, y-axis, the line y = x, the line y = -x, or none of these.

Substituting (a, b) into the equation yields . Check to see if each test produces an equation equivalent to .

x-axis	
	
y-axis	
	
y = x	
	
y = -x		

Therefore, the graph of  is symmetric with	
respect to the y-axis. A sketch of the graph verifies
the algebraic tests.

	

Example 3
Determine whether the graph of |y| = 4 - 2|x| is symmetric with respect to the x-axis, the
y-axis, both, or neither. Use the information about the equation’s symmetry to graph the relation.

Substituting (a, b) into the equation yields . Check to see if each test produces an equation equivalent to .

x-axis	
	
y-axis		

Therefore, the graph of  is symmetric with respect to both the x-axis and the y-axis.

To graph the relation, let us first consider ordered pairs where x ≥ 0 and y ≥ 0. The relation  contains the same points as y = 4 - 2x in the first quadrant.

Therefore, in the first quadrant, the graph	
of  is the same as the graph
of y = 4 - 2x.

Since the graph is symmetric with respect	
to the x-axis, every point in the first
quadrant has a corresponding point in the
fourth quadrant.

Since the graph is symmetric with respect	
to the y-axis, every point in the first and
fourth quadrants has a corresponding
point on the other side of the y-axis.

	

Example 4
CRYSTALLOGRAPHY  A crystallographer sketches	
the outline of a crystal having x- and y-axis symmetry
on a graph. He uses the piecewise function
 to model the first quadrant
portion of the cross-section. Write piecewise equations
for the remaining sides.

Since the graph has x-axis symmetry, substitute (x, -y) into the original equation to produce the equation for the fourth quadrant portion.

The equation models the
fourth quadrant portion of the cross section.
Since the graph has y-axis symmetry, substitute (-x, y) into the first and fourth quadrant equations to produce the equations for the second and third quadrants.

The equations and model the cross the second and third quadrant potions of the cross section respectively.