### E3 (adv) - write the equations of circles and ellipses in transformational form and as mapping rules to visualize and sketch graphs.

Lindsay

**Definitions:**

**Circle:** a single, closed curved line

**Ellipse:** a single, closed curved line that has one axis longer than the other (x-axis or y-axis) *Example below has a longer y-axis*

**Writing an equation for a circle:** To write an equation for a circle, it first needs to be put on a graph. Let's make it easy and have the centre of the circle at the origin of the graph (0,0).

The equation of a circle looks like this: **[1/r (x-h)]^2 + [1/r (y-k)]^2 = 1 **This is called **Transformational Form. **The **r** stands for **radius**, **h** is the **horizontal translation** and **k** is the **vertical translation**. **h** and **k** is also the **centre of the circle**.

The equation for the circle above circle is: **[1/5 (x-0)]^2 + [1/5 (y-0)]^2 = 1**

If we only had the equation and not the diagram of the circle, we would need a **mapping rule** to make the diagram. A mapping rule looks like this:

**(x,y) -> (x+h, ay+k) **Notice how the **h** and **k** are now positive? This becomes important later. The **a** stands for the **stretch**, but it not needed here.

The mapping rule for the above circle is: **(x,y) -> (x+0, y+0)**

Let's make it harder and change the circle on the graph:

The equation fo this circle is: **[1/5 (x-0)]^2 + [1/5 (y-3)]^2 =1 **All we did with this circle was **move it vertically by +3**. This **+3 is equal to k**. This is what happened: **[1/5 (x-(0)]^2 + [1/5 (y-(+3)]^2 =1** We did not move the circle horizontally from our last circle so it stays at 0.

The mapping rule now is: **(x,y) -> (x+0, y+3)** In the mapping rule, the **h and k are positive**, unlike the equation where they are negative. This is to show that the circle moved +3 on the y-axis. It makes it easier to draw.

**Writing an equation for an ellipse: **The equation for an ellipse is the same as an equation for a circle except we now have to use a **stretch**. We'll start off with an ellipse with a centre at the origin (0,0).

The equation of an ellipse is: **[1/a (x-h)]^2 + [1/b (y-k)]^2 =1** This is also called **transformational form**. In this equation, the only difference from the circle equation is the **a** and **b**. Both **a and b stand for stretch**. **a** is usually the longer stretch but it does not matter.

The equation for the above ellipse is: **[1/3 (x-0)]^2 + [1/6 (y-0)]^2 =1**

The mapping rule for the above ellipse is: **(x,y) -> (x+0, 2y+0)** Because this ellipse has a longer y-axis, the stretch goes with the y in the mapping rule. In this ellipse, the y-axis is twice the length of the x-axis, giving it a stretch of 2.

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