_{Up Learn – A Level Maths (edexcel) – Coordinate Geometry II}

_{Up Learn – A Level Maths (edexcel) – Coordinate Geometry II}

**Parametric Function Tables**

**We can understand how parametric equations work by looking at parametric function tables.**

### More videos on Parametric Equations: Summary:

The Coordinates Given by Parametric Equations

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## Coordinate Geometry II

2. What are Parametric Equations?

3. Parametric Function Tables

4. The Coordinates Given by Parametric Equations

5. Sketching the Curves of Parametric Equations

6. Sketching Curves within a Restricted Domain

7. What is a Parameter?

8. Turning Parametric Equations into a Cartesian Equation

9. Taking Shortcuts when finding Cartesian Equations

10. Turning Cartesian Equations into Parametric Equations

11. Trigonometric Parametric Equations

12. The Problem with Trigonometric Parametric Equations

13. Converting When the Trig Functions are the Same

14. Converting When the Trig Functions are Reciprocals

15. Using the Pythagorean Identity

16. A Faster Way to Use the Pythagorean Identity

17. Using the Other Pythagorean Identities

18. The Secret Power of the Reciprocal Identities Part 1

19. The Secret Power of the Reciprocal Identities Part 2

20. Parametric Equations We Can Convert So Far

21. Multiple Trig Functions in One Parametric Equation

22. Using the Double Angle Identities

23. Using the Compound Angle Identities

24. The Domain and Range of Parametric Equations

25. Finding Unknown Coordinates

26. Finding Points of Intersection in Parametric Form

27. Points of Intersection with Parametric Trig Equations

2. The z-axis

3. 3D Coordinates

4. Plotting 3D Coordinates

5. Rotating the Euclidean Space

6. Pythagoras in 3D – Part 1

7. Pythagoras in 3D – Part 2

8. Distance from the Origin

9. Distance Between Two Points

Last time, we saw that there are two ways of representing curves…

In cartesian form, and in…

In cartesian form, and in parametric form…

For example, which of these curves have been represented in parametric form?

[y=2-ex]

[y=x2+4x+5]

[y=e3ln(x)]

[y=3-1x]

[y=1×2+4x+4]

[y=(x3)6+2(x3)2, x>0]

These curves have been represented in cartesian form, with one, cartesian equation…

But these have been represented in parametric form, with two, parametric equations…

And now, to see how exactly parametric form works… how it’s able to look so different, yet produce the exact same curve as Cartesian Form, we’ll start by constructing a function table…

Now, when we’ve constructed function tables in the past, we’ve always done it for cartesian equations…

Like this one [y=2×2+12x+11]

And then, we’ve written values for x here, and related values for y here…

And, we’ve said that, certainly for functions at least, one other way of thinking about these x values is as inputs, and another way of thinking about these y values is as outputs…

But now that we’re working with parametric equations…

Like these [x=t-3 and y=2t2-7]

We have to construct our table slightly differently… and again, thinking about these equations as functions, with inputs and outputs, helps.

First, when we do that, the input is no longer represented by x…

The input in both functions is represented by…

The input in both functions is represented by t…

So, we start with a column for t…

Then, we need to find the outputs of two functions…

We need to find the outputs of these two functions…

First, x represents the output of this function…

And second, y represents the output of this function…

So, we put a column for x here, and a column for y here…

Finally, we can fill our table with values!

First, we need to pick some inputs…

Any will do, but integers around 0 are a good place to start, since they’re easy to work with […]

[maybe we should do something to make the first column look distinct from the two outputs]

Now, when t is equal to -3, x will have a value of…

When t is equal to -3, x will have a value of -6.

Whereas y will have a value of…

y will have a value of 11.

Next, when t is equal to -2, the values of x and y will be…

When t is equal to -2, the values of x and y will be -5 and 1.

And so now, fill in the rest of this function table…

With the values for x and y filled in, the completed function table looks like this.

So next, here’s a function table for another pair of parametric equations… [relatively simple]

Fill in the missing gaps in the table…

This time, the completed function table looks like this.

So, to sum up, when we’re working in parametric form, the input is represented by…

When we’re working in parametric form, the input is no longer represented by x – but by t instead.

And so, when drawing up a function table for parametric equations, we need…

We need a column for t first, and then a column for x and y, which now both represent…

Which now both represent outputs…