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Learning intentions:
In this section we will examine:
- Circles and semicircles and their Cartesian equations.
- How to graph circles and semicircles from a given equation.
- Transformations of circles.
Circles
Circles are not functions - they fail the vertical line test.
Consider a circle with a centre at the origin (0, 0) and a radius of r units. The distance from the origin to the circles' radius, in Cartesian form, is given by Pythagoras' theorem:
Figure 1 - A circle, with radius r, defined by the Cartesian equation: x2 + y2 = r2
Centre-radius form of a circle
The centre-radius equation of a circle is:
- The centre of the circle is located at (h, k).
- The circle has a radius of √r2 = r.
All circle graphs have the same shape, they are just transformed (dilated and translated) according to the values of the parameters r, h and k.
General form of a circle
Circles can also be written in expanded form:
When this is the case, you need to complete the square with respect to x and y to get it into the centre-radius form - then you can easily determine the radius and centre of the circle.
Looking ahead: Domain of a relation
In the near future we will examine the domain of a relation. The domain is the set of all of the x-coordinates (or first elements of an order pair). When we consider domains of relations they can be maximal/implied or they can be restricted.
- For the circle relation, the maximal domain is restricted to radius of the circle which can also be translated according to the value of h. In general, the domain of a circle is given by: x ∈ [h - r, h + r]
Looking ahead: Range of a relation
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 circle relation, the range is restricted to radius of the circle which can also be translated according to the value of k. In general, the domain of a circle is given by: x ∈ [k - r, k + r]
Transformations of the circle graph
When considering transformations of the circle graph, it is easiest to have the equation in the following form:
We can consider the effects of each parameter (r, h and k) on the circle graph.
- r will change the radius of the circle
- 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 a circle.
- Please use the sliders to adjust the parameters and observe the transformations.
Coming soon!
Graphing circles
When graphing circles you must label:
- Any axial intercepts (if they exist) with their coordinates.
- The coordinates of the centre of the circle.
- It is also good practice to label the four extrema point of the circle.
Before graphing circles, always make sure it is in the following form:
6E - VIDEO EXAMPLE 1:
Graph the following circle and state the maximal domain and range:
To find the x-intercepts let y = 0 and solve for x. This will involve solving a quadratic. You can also use the general quadratic formula.
To find the y-intercepts let x = 0 and solve for y. This will involve solving a quadratic. You can also use the general quadratic formula.
6E - VIDEO EXAMPLE 2:
Graph the following circle and state the maximal domain and range:
6E - VIDEO EXAMPLE 3:
Graph the following circle and state the maximal domain and range:
Semicircles
Semicircles are functions.
Consider a circle with the equation x2 + y2 = r2. We can solve for y using inverse operations:
A semicircle is found by taking either the positive or negative statement:
Figure 2 - Two semicircles combine to give a circle.
General equation of a circle
The general equation of a semicircle is:
- The middle of the semicircle is located at (h, k).
- The semicircle has a radius of √r2 = r.
- a is generally 1 or -1; however, other dilations are possible.
- If a is positive the top of the circle is present (concave down).
- If a is negative the bottom of the circle is present (concave up).
All semicircle graphs have the same shape, they are just transformed (dilated and translated) according to the values of the parameters a, r, h and k.
Graphing semicircles
When graphing circles you must label:
- Any axial intercepts (if they exist) with their coordinates.
- The coordinates of the middle of the circle.
- It is also good practice to label the three extrema point of the circle.
Before graphing semicircles, always make sure it is in the following form:
6E - VIDEO EXAMPLE 4:
Graph the following semicircle and state the maximal domain and range:
VIDEO HERE
6E - VIDEO EXAMPLE 5:
Graph the following semicircle and state the maximal domain and range:
VIDEO HERE
6E - VIDEO EXAMPLE 6:
What is the equation of the following semicircle?
VIDEO HERE
Success criteria:
You will be successful if you can:
- Sketch circles and semicircles from their Cartesian equations.
- Label axial intercepts (if they exist), locate the centre of the circle and determine the radius.
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