How to think about co-ordinate geometry (Part II: curves, tangents and normals)

This is part two of a three-part series about co-ordinate geometry. In part I last week, I went into tedious detail about the equation of a line. This week, I’m going to take it a bit further and go into curves. Next week, you get to see circles.

So, what is a curve?

You don’t really need a technical definition, and it probably wouldn’t help you even if I could provide one. I’m going to give you a loose definition that a curve is anything you can draw. Obviously, that’s a pretty wide-ranging definition, and there’s only a limited subset of all of the possible curves you need to care about for C1.

In most of A-level, you only care about functions, which have the nice quality that they never back-track: for any value of $x$ you can think of, if you draw a vertical line through that value, it crosses the curve once. Or nonce.

Every curve has a (possibly very complicated) equation in the form $y=f(x)$, where $f(x)$ is some jumble of $x$s and numbers. Just like with the straight line, you can tell whether a point is on the line by checking the two sides of the equation: replace the $y$ with the $y$-coordinate and the $x$s with the $x$-coordinate and make sure the two sides give you the same answer.

A curve also (as far as you’re concerned) has a derivative, $\frac{dy}{dx} = f’(x)$, which you get by differentiating the jumble of $x$s. This tells you how steep the curve is at any given point: you just throw in the value of $x$ and see what comes out.

Curves are objects that often have names (silly names like $C$) — I find it helpful to think of them like Top Trumps cards with categories like “Equation of curve”, “Equation of derivative”, “Name”, “$y$-intercept”, “Solutions”, "Turning points" and so on. You can even draw out the card if it helps…

(A particularly useful thing to note: if the gradient is 0, the curve is temporarily flat; this is known as a turning point, or a stationary point, or an extremum, or a local maximum or minimum, depending on how awkward they want to be.)

What’s a tangent?

Tangent — as an adjective — means ‘touching’. As a noun, in maths, it means ‘the (unique) straight line that touches the curve at a given point, and has the same gradient as the curve there.’

You can draw it (at least approximately) without too much effort: you just put a ruler down so it touches your curve and have at it with a pencil. It’s always worth doing this (assuming you can sketch it), just to get an idea of what it ought to look like — being able to say “it needs to be a steep line” gives you a clue about the gradient of it.

If you want to find the equation of a tangent to a curve at a given point — which is just a straight line, remember — you need two things: a gradient and a point on the line. Like you do with all straight lines.

You get the gradient by looking at the derivative of the curve and putting your $x$-value in. The number that comes out is the gradient of your line.

You get a point on the line by using the equation of the curve (assuming you weren’t given both coordinates to start with). Usually, you’ll just throw in the $x$-value you’re given, but they may be awkward and give you the $y$-value instead.

Once you have the gradient and a point on the line, you’re away: you do the $(y-y_0) = m(x-x_0)$ dance again and there you have it.

What’s a normal?

A normal is simply the line at right angles to the tangent to a curve at a given point. (The tangent touches; the normal is at ninety bad-degrees.)

Finding the equation of a normal isn’t too rough: if you can find the gradient of the curve, $m$, (using the derivative, just like before), you can find the perpendicular gradient, like you looked at last week, by working out $-1/m$. You can find a point on the line just like before, and now you have all you need. Boom: throw it in the formula and there’s your straight line.

How about working backwards?

The most awkward questions are ones where they ask for points where the tangents are parallel, or where the normal intersects the curve again. These are awkward only because they require you to do a bit of thinking rather than following a simple recipe. Got that brain of yours switched on? Good-o.

This kind of question is all about making two things equal. For example, if you want to find a point where the gradient of the tangent is $m$, you need to solve the equation $f’(x)=m$ — where $f’(x)$ is the derivative you worked out earlier.

To find where a line intersects a curve, you probably want simultaneous equations: you’ve got two equations ($y=f(x)$ for the curve, and an equation for the line), both of which need to be true, which is a great big sign saying “Simultaneous Equations ahoy!”

Next week, I’ll show you how to apply this kind of reasoning to circles. Won’t that be fun?

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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