SECTION 11.3 The Hyperbola and Functions Defined by Radicals 653
OBJECTIVE 3 Identify conic sections by their equations.Rewriting a second-
degree equation in one of the forms given for ellipses, hyperbolas, circles, or parabolas
makes it possible to identify the graph of the equation.
Identifying the Graphs of EquationsIdentify the graph of each equation.
(a)
Both variables are squared, so the graph is either an ellipse or a hyperbola. (This
situation also occurs for a circle, which is a special case of an ellipse.) Rewrite the
equation so that the - and -terms are on one side of the equation and 1 is on the
other.
Subtract.Divide by 108.The graph of this equation is a hyperbola.
(b)
Only one of the two variables, x, is squared, so this is the vertical parabola
(c)
Write the variable terms on the same side of the equation.
AddThe graph of this equation is a circle with center at the origin and radius 3.
NOW TRYOBJECTIVE 4 Graph certain square root functions.Recall from the vertical
line test that no vertical line will intersect the graph of a function in more than one
point. Thus, the graphs of horizontal parabolas, all circles and ellipses, and most
hyperbolas discussed in this chapter do not satisfy the conditions of a function. How-
ever, by considering only a part of each graph, we have the graph of a function, as seen
in FIGURE 23.
x^2 +y^2 = 9 y 2.
x^2 = 9 - y^2
y=x^2 + 3.
x^2 = y- 3
x^2
12
-
y^2
9
= 1
9 x^2 - 12 y^2 = 108 12 y 2
x^2 y^2
9 x^2 = 108 + 12 y^2
EXAMPLE 3
x
0yxy0
x
0yx
0yxy0(a) (b) (c) (d) (e)
FIGURE 23In parts (a) – (d) of FIGURE 23, the top portion of a conic section is shown
(parabola, circle, ellipse, and hyperbola, respectively). In part (e), the top two por-
tions of a hyperbola are shown. In each case, the graph is that of a function since the
graph satisfies the conditions of the vertical line test.
In Sections 8.1 and 11.1,we observed the square root function defined by
To find equations for the types of graphs shown in FIGURE 23, we extend
its definition.
ƒ 1 x 2 = 2 x.
NOW TRY
EXERCISE 3
Identify the graph of each
equation.
(a)
(b)
(c) 3 x^2 +y^2 = 4
y- 2 x^2 = 8y^2 - 10 =-x^2NOW TRY ANSWERS
- (a)circle (b)parabola
(c)ellipse