By the end of this lab you should:
1.) Be familar with the important features of hyperbolas, namely their vertices
2.) Understand the connection between the equation of thehyperbola and its
3.) Be able to draw hyperbolas whose equations are in standard form.
4.) Be able to draw contour diagrams of some important quadratic functions
Calculus is about functions. Some basic examples are linear functions like f(x,
3x + 4y and quadratic functions like g (x, y) = 3x2 + 4y2 or h(x, y, z) = 3x2 +
4y2 − z2.
Calculus is powerful, because it often reduces understanding a complicated
understanding a linear or quadratic one. Still, we have to understand the linear
quadratic functions. In order to understand a quadratic function like g(x, y) =
3x2 − 4y2,
we have to understand its levels. These are the curves g (x, y) = c, and they are
In this computer lab, we will review hyperbolas. We will need to be able to draw
curves so that we can draw level curves of quadratic functions. It will turn out
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
see how the shape of the hyperbola is related to the terms of the equation, and
to draw the graphs of the equations .
As we all know, the equation x2 + y2 − 1 = 0 describes a circle in the xy-plane.
general, quadratic equations of the form
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
describe plane curves known as conic sections. For different choices of the
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.
The graph of the equation x2/a2−y2/b2 = 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
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
are called vertices, and they lie on the x-axis, if the hyperbola is in standard
The hyperbola also has two asymptotes. The equations for the asymptotes are
by taking the quadratic function x2/a2 − y2/b2, setting it equal to zero,
factoring it, and
setting each factor equal to zero. The two equations for the asymptotes that we
x/a + y/b = 0
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
included them in the figure.
(a) Plot x2/a2−y2 = 1 for a = 1, 3, 5, 7. We suggest you use a do loop of the
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 .
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 x2 − y2/b2 = 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
and whose vertices are located at (−6, 0), (6, 0).
Attach a printout of the graph of your hyperbola.
Question 3. Plot x2 − y2 = 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,
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
Drawing Hyperbolas by Hand
To draw a hyperbola, first plot the vertices, then draw the asymptotes, then
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
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
the easier it will be to describe it.)
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