A variable, e.g. x, represents an unknown number.

A function, e.g. f(x) (which is read as “F of x”, or “Some function of x”), is like a ‘super variable, representing not just an unknown number, but a whole expression, such as 2x2 + 5x + 8, or 3x, or 4x – 2, or x2.

These functions are usually plotted as algebra graphs, and when we talk about ‘transforming’ them, we mean moving all the points up, down, left or right, or reflecting the function in the x or y axis, or squishing all the points closer together, or stretching them out.

Without redrawing the function, we can describe the transformation mathematically by rewriting the y = f(x) (which is sometimes referred to as the graph of fx) like this:

y = f(x + a) + m  Meaning subtract a from all of the original x coordinates, and add m to all of the original y coordinates

y = nf(bx)  Meaning divide all of the original x coordinates by b, and multiply all of the original y coordinates by n.

It can be thought of in general as “Whatever is inside the brackets, do the opposite to the new x-coordinates, and whatever is outside the bracket, do exactly what it says to all of the new y-coordinates.”

y = f(-x)  Represents a reflection in the y-axis, i.e. all new x-coordinates have their signs flipped

y = -f(x)  Represents a reflection in the x-axis, i.e. all new y-coordinates have their signs flipped

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