# The Coordinates Given by Parametric Equations

When you plot parametric equations, the x and y axes both represent outputs.

### More videos on Parametric Equations: Summary:

Parametric Equations: Summary

Parametric Function Tables

The Coordinates Given by Parametric Equations

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Up Learn – A Level Maths (edexcel)

## Coordinate Geometry II

1. Introduction to Parametric Equations
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
1. Introduction to 3D Coordinates
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 how to construct a function table for parametric equations.

For example, find the missing values in this function table [x=t3 and y=4-2t]

To find the missing values of x, we use this equation…

And to find the missing values of y, we use this equation…

Meaning our completed function table looks like this

And so now, with a completed function table, we have a bunch of x and y coordinates, which we can plot like this…

[highlight those two columns or fade out t column or something, to make it reminiscent of a normal

function table] [table to side of plane, plot points one by one]

And then, we join up the points…

So, these parametric equations represent this curve!

Now, x and y coordinates can always be represented as points on the cartesian plane

Meaning, once we have x and y coordinates, it doesn’t matter whether we found them using cartesian or parametric equations…we still plot them in exactly the same way.

However, when they’re generated using parametric equations, we have to think about the relationship between x and y coordinates a little differently.

For example, when generated by a single Cartesian Equation, we said that the x and y values can often

be thought of as…

The x value can often be thought of as an input to a function, and the y value as the function’s output.

But, when generated by parametric equations, both the x and y values are…

When generated by parametric equations, both the x and y values are outputs of functions, and a new value, the parameter t, is the only input…

And so, in parametric form, we no longer plot the input on either axis – our parameter t is like a secret input, hiding behind the curve, not plotted but still pulling all the strings…

To sum up, there are some crucial differences between cartesian form and parametric form…

With cartesian equations, we represent a curve with one equation…

Then, we plot its inputs on the horizontal axis, and its outputs on the vertical axis, like this […]

But with parametric equations, we need two equations…

Then, we plot the outputs of the first on the horizontal axis, and the outputs of the second on the vertical axis, like this […]

Leaving our inputs unplotted – hiding behind the graph, secretly pulling all the strings…