College Algebra 9th txtbk

SECTION 2–1 Cartesian Coordinate Systems 117
5
y
x 0 2 4
5
y x
5
x
y 0 2 4
5
Z Figure 9
MATCHED PROBLEM 5 Test the equation y x for symmetry and sketch its graph.
EXAMPLE 6
Using Symmetry as an Aid to Graphing
Test the equation y 2 x 2 4 for symmetry and sketch its graph.
SOLUTION
Since (x) 2 x 2 and (y) 2 y 2 , the equation y 2 x 2 4 will be unchanged if x is
replaced with x or if y is replaced with y. So the graph is symmetric with respect to
the y axis, the x axis, and the origin. We need to make a careful sketch in only the first
quadrant, reflect this graph through the y axis, and then reflect everything through the x axis.
To find quadrant I solutions, we solve the equation for either y in terms of x or x in terms
of y. We choose to solve for y.
y 2 x 2 4
y 2 x 2 4
y 2x 2 4
To obtain the quadrant I portion of the graph, we sketch y 2x 2 4 for x 0, 1, 2, . . . .
The final graph is shown in Figure 10.
x 0 1 2 3 4
y 2 15 2.2 18 2.8 113 3.6 120 4.5
5
y
y 2 x 2 4
5
(3, √13) (4, √20)
(2, √8)
(0, 2) (1, √5)
x
5
5
Z Figure 10
MATCHED PROBLEM 6
Test the equation 2y 2 x 2 2 for symmetry and sketch its graph.