Parabola, Hyperbola, Circle, and Ellipse

Graph of a single equation

Before solving a quadratic system graphically, the method of graphing a single quadratic equation in two variables must be clearly understood.

Example 1. Construct the graph of y^2 = 4x.

Solving the equation for y in terms of x, y = \pm 2\sqrt{x}. We now assign values to x and compute the approximate corresponding values of y:

x =	9	4	3	2	1	0	-1	any negative x
y =	\pm6	\pm4	\pm3.5	\pm2.8	\pm2	0	\pm2\sqrt{-1}	imaginary

Plotting the points that correspond to the real numbers in the table, and drawing the curve they determine, we obtain the graph below. Here y is a function of x, and the y-axis is the function axis. The curve is a parabola, like those of Chapter XXII.

y^2 = 4x

The graph of any equation of the form y^2 = ax is a parabola.

Example 2. Graph the equation xy = 6.

Solving for y, y = \dfrac6x. Assigning values to x and computing the values of y:

x =	-6	-5	-4	-3	-2	-1	- \frac{3}{4}	\frac{3}{4}	1	2	3	4	5	6	8
y =	-1.0	-1.2	-1.5	-2.0	-3.0	-6.0	-8	8	6.0	3.0	2.0	1.5	1.2	1.0	0.75

We obtain the two-branched curve below. The curve does not touch either axis. When x increases numerically, y decreases; as x decreases from 1 toward 0, y becomes indefinitely great. Similarly, when x has a numerically great negative value, y has a numerically small negative value, and vice versa. The curve is called a hyperbola.

xy = 6

The graph of any equation of the form xy = K is a hyperbola. If K is positive, one branch of the curve lies in the first quadrant and the other in the third. If K is negative, one branch lies in the second quadrant and the other in the fourth.

Example 3. Graph the equation x^2 + y^2 = 25.

Solving for y, y = \pm\sqrt{25 - x^2}:

x =	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
y =	\pm\sqrt{-11}	0	\pm3	\pm4	\pm4.58	\pm4.90	\pm5	\pm4.90	\pm4.58	\pm4	\pm3	0	\pm\sqrt{-11}

For values of x numerically greater than 5, y is imaginary. The points that correspond to the real numbers in the table lie on the circle below. Its center is at the origin and its radius is 5.

x^2 + y^2 = 25

Further, the graph of any equation of the form x^2 + y^2 = r^2 is a circle whose radius is r. This follows from the right triangle PKO: if P is any point on the circle, OK is the x-distance of P, PK its y-distance, and OP the radius. Now \overline{OK}^2 + \overline{PK}^2 = \overline{OP}^2; that is, x^2 + y^2 = r^2. So the graphs of x^2 + y^2 = 9 and x^2 + y^2 = 8 are circles about the origin with radii 3 and \sqrt8. To graph x^2 + y^2 = r^2 you may simply use compasses, with the origin as center and the proper radius.

Example 4. Graph the equation 4x^2 + 9y^2 = 36.

Solving for y, y = \pm\frac23\sqrt{9 - x^2}:

x =	-4	-3	-2	-1	0	1	2	3	4
y =	\pm\frac23\sqrt{-7}	0	\pm1.49	\pm1.89	\pm2	\pm1.89	\pm1.49	0	\pm\frac23\sqrt{-7}

For values of x numerically greater than 3, y is imaginary. The curve is called an ellipse.

4x^2 + 9y^2 = 36

The graph of any equation of the form ax^2 + by^2 = c, in which a and b are unequal (and positive), is an ellipse. If a and b have opposite signs, as in x^2 - y^2 = 16, the graph is a hyperbola.

Reading a curve from its equation

Form	Curve
y^2 = ax or x^2 = ay	parabola
x^2 + y^2 = r^2	circle of radius r
ax^2 + by^2 = c, a \ne b, both positive	ellipse
xy = K, or ax^2 - by^2 = c	hyperbola

Where a curve crosses the x-axis, y = 0; where it crosses the y-axis, x = 0.

Check with the calculator. The calculator's Graph tab draws these curves directly: type the equation as it stands, for example x^2 + y^2 = 25 or xy = 6, or press the Graph button beside an equation in the lesson. Draw your own graph first; then compare.

Try one:

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Quick check

What curve is the graph of x^2 + y^2 = 49, and what is its radius?

These three curves, the ellipse, the hyperbola, and the parabola, were first studied by the Greeks, who proved that they are the sections which one obtains by cutting a cone by a plane. Not for hundreds of years afterwards did any one imagine that these curves actually appeared in nature, for the Greeks regarded them merely as geometrical figures, and not at all as curves that had anything to do with our everyday life. One of the most important discoveries of astronomy was made by Kepler (1571–1630), who showed that the earth revolves around the sun in an ellipse, and stated the laws that govern the motion. Those comets that return to our field of vision periodically also have elliptic orbits, while those that appear once, never to be seen again, describe parabolic or hyperbolic paths. The path of a ball thrown in the air is also a parabola.

Parabola, Hyperbola, Circle, and Ellipse

Graph of a single equation

Reading a curve from its equation

Exercises