**Final **

**by **

**Julie Anne Laycock**

**Part 1: Consider the equation xy = ax + by + c**

Let's first look at what happens when we set **a** and **b** at 1 and change the **c** value. In this case, the **c** is 1,2,3, and 4. We can see we get a set of hyperbolas. As **c** increases the hyperbolas move up and down the graph by each unit. We can see the x and y intercepts change by one unit as the **c** changes by one unit.

Now let's look at what happens when **c** is changed to a negative. The **c** values here are 1, -1, 2, and -2. When the **c** is -2 we can see the hyperbola has flipped the other direction. And it looks like when **c** is -1 the hyperbolas connect. Let's look at this a bit closer.

We can see when we set the equations (y - a) and (y - b) equal to zero we get the horizontal and the vertical asymptotes for the hyperbolas. Also when we multiply the equations together and set it equal to **n**. The animation shows **n** varying from -5 to 5. We can see that as **c** gets closer and closer to 0 the hyperbolas are approaching the asymptotes. When **c** is 0 they hyperbolas merge onto the asymptotes and when **c** is less than 0 the the hyperbolas flip the other direction. As **c** gets bigger the hyperbolas move further and further away from the asymptotes. Also, as **c** gets smaller and smaller the hyperbolas also are moving further and further away from the asymptotes.

**Part 2: GSP sketch and script tool for constructing a rhombus. **

**Rhombus given an angle and a side.**

Script tool: Click Here

**Rhombus given an angle and a diagonal.**

Script tool: Click Here

**Rhombus given an altitude and a diagonal.**

Script tool: Click Here

**Part 3: Write-up of an additional item from the assignments. **

Explore :

In the graph of we get a parabola with the vertex at the origin. Let's look as what happens to the graph when we change the value of **a**. In the image on the left a is set at 1, -1, 2, and -2. We can see when the **a** is positive the parabola is facing up and when **a** is negative the parabola is facing down. Also, we notice as the absolute value of each number increases the parabola is stretching. Let's look at this a little closer.

In this animation the **a** is varying from 1 to 30. We can see as the the number gets bigger and bigger the parabola is being stretched and it's getting closer and closer to the y axis. Next, let's look as what will make the parabola wider.

In these graphs, **a** is set at 1, 0.5, 0.25, -0.25, -0.5. We can see when the absolute value of a gets smaller and closer and closer to 0 the parabola is widening.

In this animation, the **a** is varying from 0 to 1. We can see as a gets closer and closer to 0 the parabola is widening until it becomes the x axis when a = 0.