02.21hyperbola

The Hyperbola

… {pronounced hi-PER—bola}

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 = xn.

• 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

… … $y = x^{-1}$ can be written as $y = \dfrac{1}{x}$

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 but never reaches zero (from above ie from the positive side of 0).
• In symbols, as $x \rightarrow +\infty, \; y \rightarrow 0^+$
• The line $y = 0$ (the x-axis) is called an asymptote because the y-value approaches but never reaches 0.
• At the negative end, the above table would be the same, but with negative values.
• So as the value of x gets very large in the negative direction, the value of y approaches but never reaches zero (from below ie from the negative side of 0)
• In symbols, as $x \rightarrow -\infty, \; y \rightarrow 0^-$

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 $x \rightarrow 0^+, \; y \rightarrow +\infty$
• Therefore $x = 0$ is also an asymptote because the x-value approaches but never reaches 0.
• Again, for negative values of x, the above table would be the same except every value would be negative.
• So as the value of x approaches 0 from the negative side, the value of y decreases to negative infinity.
• In symbols, as $x \rightarrow 0^-, \; y \rightarrow -\infty$

Graph of a Standard Hyperbola

Now plot some of these points and sketch the graph. Clearly draw in and label the asymptotes.

Graph of Standard Hyperbola: $y = \dfrac{1}{x}$ The asymptotes are along the axis, so the equations of the asymptotes are $x = 0 \text{ and } y = 0$

The domain is all Real values of x, except 0.

• Domain: $x \in R \backslash \{ 0 \}$
• Range: $y \in R \backslash \{ 0 \}$

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.

Example 1 Example 2 Note

• This is the standard hyperbola with the following transformations:
• reflected over the x-axis (in the y direction)
• dilated by a factor of 0.5 from the y-axis (in the x direction)
• translated 1.5 units to the right
• translated 2 units up

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

Example:

… … Find the equation of the asymptotes for $y = \dfrac{2}{3x-4} - 2$

Solution:

vertical asymptote: set denominator to zero and solve for x

… … $3x - 4 = 0$

… … $x = \dfrac {4}{3}$

horizontal asymptote: replace fraction with zero and solve for y

… … $y = 0 - 2$

… … $y = -2$ .