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.

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!”