# 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 = 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

… … $y = x^{-1}$ 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

… … $y = 0 - 2$

… … $y = -2$

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