02Chyperbola

toc = The Hyperbola =

{pronounced hi-PERB-ola}

A hyperbola is one of a group of shapes called conic sections (not in course). A hyperbola has practical applications in Physics and Astronomy (not in course).

Power Functions
A power function has the form y = x n.
 * For n = 1, we get a linear graph,
 * For n = 2 we get a parabola,
 * For n = 3 we get a cubic.
 * When n = –1, we get a graph called a ** hyperbola **.

The Standard Hyperbola
math y = x^{-1} \text{ can be written as } y = \dfrac{1}{x} \qquad. math

If we consider a table of values for this function

We can see that as the value of x gets very large, the value of y approaches zero but never reaches zero. In symbols, as x → ¥, y → 0

The line y = 0 (the x-axis) is called an asymptote because the y-value approaches but never reaches 0.

For the same function, if we consider x-values between 0 and 1, we get:

Looking at the table from right to left, we see that as the value of x decreases from 1 to 0, the value of y increases to infinity. At x = 0, y is undefined. In symbols, as x → 0, y → ¥

Therefore x = 0 is also an asymptote because the x-value approaches but never reaches 0.

Notice that for negative values of x, our tables would be the same except every value would be negative.

Graph of a Standard Hyperbola
Now plot some of these points and sketch the graph. Clearly draw in and label the asymptotes. math y = \dfrac{1}{x} \qquad. math

The asymptotes are along the axis, so the equations of the asymptotes are x = 0 and y = 0

The domain of is all Real values of x, except 0. Domain: x Î R\{0} Range: y Î R\{0}

There are no stationary points and no intercepts.

The graph passes through the points (1, 1) and (–1, –1)

Transformations on the Hyperbola
We can apply the standard transformations such as dilations, translations and reflections when sketching a hyperbola. media type="custom" key="7959718" media type="custom" key="7959720" ... media type="custom" key="7959724" ... media type="custom" key="11178348" ... media type="custom" key="11178358" ... media type="custom" key="11178360"

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 * Note **
 * more complicated rational functions can be simplified by performing long division.
 * the vertical asymptote can always be found by setting the denominator to 0 and solving for x.
 * the horizontal asymptote can be found in the simplified form by replacing the fraction with zero and solving for y

Find the equation of the asymptotes for the following graph; math . \qquad y = \dfrac{2}{3x-4} - 2 \qquad. math
 * Example: **


 * Solution: **

math . \qquad \qquad 3x - 4 = 0 \qquad. \\ . \\ . \qquad \qquad x = \dfrac {4}{3} math
 * vertical asymptote: set denominator to zero and solve for x

math . \qquad \qquad y = 0 - 2 \qquad. \\ . \\ . \qquad \qquad y = -2 math
 * horizontal asymptote: replace fraction with zero and solve for y

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