5F - Circles and semicircles
Circles:
Consider the 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: \(x^2+y^2=r^2\) ■ So far we have referred to many graphs as functions. In Section 6 we will explore the idea of a function in more detail. At this point in time, please note that circles are not considered functions as they fail the vertical line test. |
The centre-radius form of a circle: \[(x-h)^2+(y-k)^2=r^2\]
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The general form of a circle: \[x^2+ax+y^2+by+c=0\]
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Graphing circles:
When graphing circles you must include:
When graphing circles you must include:
- The axial intercepts (if they exist), labelled with their coordinates.
- The centre of the circle and a representation of the radius in some way (often by marking the extreme values).
5F - Example 1: Graphing a circle in centre-radius form
Graph the following circle and state the maximal domain and range: \[x^2+(y-2)^2=4\]
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5F - Example 1: Video solution
5F - Example 1: Practice
Question 1: ABC Question 2: ABC 5F - Example 1: Solutions
Question 1: ABC Question 2: ABC |
5F - Example 2: Graphing a circle in centre-radius form
Graph the following circle and state the maximal domain and range: \[16=(x+2)^2+(y-3)^2\]
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5F - Example 2: Video solution
5F - Example 2: Practice
Question 1: ABC Question 2: ABC 5F - Example 2: Solutions
Question 1: ABC Question 2: ABC |
5F - Example 3: Graphing a circle in general form
Graph the following circle and state the maximal domain and range: \[x^2+2x+y^2-4y-4=0\]
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5F - Example 3: Video solution
5F - Example 3: Practice
Question 1: ABC Question 2: ABC 5F - Example 3: Solutions
Question 1: ABC Question 2: ABC |
Semicircles with a base on the \(x\)-axis:
Consider a circle with the equation \(x^2+y^2=r^2\). We can solve for \(y\) by ‘undoing’ the equation: \[y=\pm\sqrt{r^2-x^2}\] A semicircle is found by taking either the positive or negative statement of the equation above. In general the equation of a semicircle is: \[y=\pm\sqrt{r^2-(x-h)^2}+k\]
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5F - Example 4: Graphing a semicircle
Graph the following semicircle and state the maximal domain and range: \[y=-\sqrt{25-x^2}\]
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5F - Example 4: Video solution
5F - Example 4: Practice
Question 1: ABC Question 2: ABC 5F - Example 4: Solutions
Question 1: ABC Question 2: ABC |
5F - Example 5: Graphing a semicircle
Graph the following semicircle and state the maximal domain and range: \[y=\sqrt{36-(x+3)^2}+7\]
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5F - Example 5: Video solution
5F - Example 5: Practice
Question 1: ABC Question 2: ABC 5F - Example 5: Solutions
Question 1: ABC Question 2: ABC |
Semicircles with a base on the \(y\)-axis:
5F - Example 6: Graphing a semicircle
Graph the following semicircle: \[x=-\sqrt{9-y^2}+3\]
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5F - Example 6: Video solution
5F - Example 6: Practice
Question 1: ABC Question 2: ABC 5F - Example 6: Solutions
Question 1: ABC Question 2: ABC |