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Learning intentions:
In this section we will examine:
- The hyperbola graph and its transformations (dilations, reflections and translations).
- How to graph the hyperbola function from a given equation.
The hyperbola graph
The hyperbola function has a general form:
- All hyperbola graphs have the same shape, they are just transformed (dilated, reflected and translated) according to the values of the parameters a, h and k.
Introduction to the hyperbola graph
Consider the following table of values with numbers and their reciprocal values:
Table 1 - Numbers and their reciprocal values.
If we plot these point on a set of axes we can see the shape of the hyperbola graph:
Draw asymptotes as a dotted line on the graph. Make sure you use a ruler!
Figure 1 - The hyperbola graph with asymptotes y = 0 and x = 0.
Asymptotes
An asymptote is a line (or curve) where the distance between the curve and the asymptote approaches zero as they tend toward infinity.
This means that the curve of the hyperbola will approach the vertical and horizontal asymptotes but never actually reach them.
Consider the hyperbola with the general equation:
The horizontal asymptote occurs at y = k.
- As x approaches ∞, y approaches k.
- As x approaches -∞, y approaches k.
The vertical asymptote occurs at x = h.
- As x approaches h (from left), y approaches ∞ (or -∞ if reflected in x-axis).
- As x approaches h (from right), y approaches ∞ (or -∞ if reflected in x-axis).
When the parameters, h and k, are changed the asymptotes undergo translation.
Looking ahead: Domain of a function
In the near future we will examine the domain of a function. The domain is the set of all of the x-coordinates (or first elements of an order pair). When we consider domains of functions they can be maximal/implied or they can be restricted.
- For the hyperbola function, the maximal domain is restricted to expression in the denominator not equalling zero. This is because you cannot divide by zero - it is undefined.
Looking ahead: Range of a function
In the near future we will examine the range of a function. The range is the set of all of the y-coordinates (or second elements of an order pair). When we consider the range of a functions they are influenced by domain and the values a function (rule) returns.
- For the hyperbola function, the range is restricted because there is no value of x which gives y = k because the term
- cannot equal zero (given that a ≠ 0).
Transformations of the hyperbola graph
When considering transformations of the hyperbola graph, it is easiest to have the equation in the following form:
We can consider the effects of each parameter (a, h and k) on the hyperbola graph.
- a will cause a vertical dilation by a factor of a from the x-axis.
- Where a < 0 the graph is reflected in the x-axis.
- h will cause a horizontal translation of h units parallel to the x-axis.
- k will cause a vertical translation of k units parallel to the y-axis.
Examining the combined effects of a, h and k
The dynamic GeoGebra worksheet illustrates the combined effect of a, h and k on the hyperbola graph.
- Please use the sliders to adjust the parameters and observe the transformations.
Coming soon!
Graphing the hyperbola function
When graphing the hyperbola function you must label:
- Any axial intercepts (if they exist) with their coordinates.
- Vertical and horizontal asymptotes with their equations.
Before graphing the square root function, always make sure it is in the following form:
To find the x-intercept let y = 0 and solve for x.
To find the y-intercept let x = 0 and solve for y.
The asymptotes exists at x = h and y = k.
6C - VIDEO EXAMPLE 1:
Graph the following hyperbola and state the maximal domain and range:
6C - VIDEO EXAMPLE 2:
Graph the following hyperbola and state the maximal domain and range:
6C - VIDEO EXAMPLE 3:
Graph the following hyperbola and state the maximal domain and range:
Success criteria:
You will be successful if you can:
- Identify and describe transformations of the hyperbola graph.
- Sketch the hyperbola graph from a given equation.
- Label axial intercepts with their coordinates and asymptotes with their equations.
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