**Goals**

By the end of this lab you should:

1.) Be familar with the important features of hyperbolas, namely their vertices
and

asymptotes.

2.) Understand the connection between the equation of the hyperbola and its
vertices

and asymptotes.

3.) Be able to draw hyperbolas whose equations are in standard form.

4.) Be able to draw contour diagrams of some important quadratic functions

**Introduction**

Calculus is about functions. Some basic examples are linear functions like f(x,
y) =

3x + 4y and quadratic functions like g (x, y) = 3x^{2} + 4y^{2} or h(x, y, z) = 3x^{2} +
4y^{2} − z^{2}.

Calculus is powerful, because it often reduces understanding a complicated
function to

understanding a linear or quadratic one. Still, we have to understand the linear
and

quadratic functions. In order to understand a quadratic function like g(x, y) =
3x^{2} − 4y^{2},

we have to understand its levels. These are the curves g (x, y) = c, and they are
examples

of hyperbolas.

In this computer lab, we will review hyperbolas. We will need to be able to draw
these

curves so that we can draw level curves of quadratic functions. It will turn out
that the

pattern of level curves around the critical points of a function of two
variables in “most”

cases looks like a family of concentric ellipses or hyperbolas .

Specifically, in this lab you will examine the equations of hyperbolas in
standard form,

see how the shape of the hyperbola is related to the terms of the equation, and
review how

to draw the graphs of the equations .

This lab will also introduce you to the MAPLE software package, which is an
extremely

powerful tool for doing mathematical calculations and graphing.

**Background**

As we all know, the equation x^{2} + y^{2} − 1 = 0 describes a circle in the xy-plane.
In

general, quadratic equations of the form

Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0

describe plane curves known as ** conic sections **. For different choices of the
constants

A,B,C,D,E, F you can get an ellipse, hyperbola, parabola, pair of lines, a
single line or

a point as the graph of the equation. These sets are
called conic sections because they are

the sets you can get if you intersect a cone with a plane.

**Hyperbolas**

The graph of the equation x^{2}/a^{2}−y^{2}/b^{2} = 1 is a
**hyperbola in standard form**. The

x and y axes are lines of symmetry for this shape. (We say a line L is a line of
symmetry

for a shape if L divides the shape into 2 congruent pieces, and the two pieces
match if we

rotate one around L.) The hyperbola has two points which are closest to the
origin; these

are called vertices, and they lie on the x-axis, if the hyperbola is in standard
form.

The hyperbola also has two **asymptotes**. The equations for the asymptotes are
gotten

by taking the quadratic function x^{2}/a^{2} − y^{2}/b^{2}, setting it equal to zero,
factoring it, and

setting each factor equal to zero. The two equations for the asymptotes that we
get are:

x/a + y/b = 0

and

x/a − y/b = 0

The figure below is a hyperbola, with vertices at (-2,0), (2,0) and with
asymptotes 4y = x,

−4y = x. So that you can see how closely the hyperbola hugs the asymptotes, we
have

included them in the figure.

**Question 1.**

(a) Plot x^{2}/a^{2}−y^{2} = 1 for a = 1, 3, 5, 7. We suggest you use a do loop of the
following

form. (The MAPLE code is explained in the glossary just before the lab)

You may also use the following commands.

These commands run faster, but they put all the plots in the same window, so you

should be sure you know which values of a go with which plots. Remember that a =
2 j−1.

**Don’t print the plots unless you need to refer to them!**

(b) Describe the changes you see in the graph as the coefficient a increases .
(Make

sure you mention how the asymptotes and vertices change. If you forget what

the asymptotes are or how to find their equations, look back in the introduction

to this lab in the material on hyperbolas.)

(c) Plot x^{2} − y^{2}/b^{2} = 1 for b = 1, 3, 5, 7.

Describe the changes you see in the graph as the coefficient b increases. (Make

sure you mention how the asymptotes and vertices change.)

Question 2. Find the equation of a hyperbola whose asymptotes have slope 1 and
−1,

and whose vertices are located at (−6, 0), (6, 0).

Attach a printout of the graph of your hyperbola.

**Question 3.** Plot x^{2} − y^{2} = c for c = −4,−2, 0, 2, 4.

Describe the changes you see in the graph as the coefficient c increases.

**Question 4.** Use the commands

to plot the level curves of f. Based on what you learned from the last question,
label those

contours where the value of f is closest to -16,-4,0,4,16. Attach your plot to
this lab. Your

MAPLE plot may have left out part of a level curve; if so, fill in the missing
part and circle

it. Notice that the asymptotes are level 0 of the function, and all the
hyperbolas have the

same asymptotes.

**Drawing Hyperbolas by Hand**

To draw a hyperbola, first plot the vertices, then draw the asymptotes, then
draw

the hyperbola so that it passes through the vertices, and approaches the
asymptotes as it

moves away from the vertices.

**Question 5.** To test what you have learned in the lab, draw the following
hyperbolas by

hand.

**Question 6.** Now draw the contour diagrams of the following
functions. Make sure your

diagram includes level 1.

**Question 7.** Write a paragraph describing the steps you
followed in drawing the diagram

in 6a). (Try to include all relevant details. The better you understand what you
are doing,

the easier it will be to describe it.)