Leave lesson

Algebra · Graphs of functions

Chapter 1 · 3

The idea

Transformations of graphs (y = f(x) + a, y = f(x + a), y = −f(x), y = f(−x))

Move a graph without plotting it again. A number added outside the f moves the curve up or down; a number added inside the bracket moves it sideways, the opposite way to its sign; a minus sign outside reflects it in the x-axis and a minus sign inside reflects it in the y-axis. Track the key points, describe each change fully, combine two changes, write the new equation, and use the same four rules on the sine and cosine waves.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Algebra · Graphs of functions

Transformations of graphs (y = f(x) + a, y = f(x + a), y = −f(x), y = f(−x))

Move a graph without plotting it again. A number added outside the f moves the curve up or down; a number added inside the bracket moves it sideways, the opposite way to its sign; a minus sign outside reflects it in the x-axis and a minus sign inside reflects it in the y-axis. Track the key points, describe each change fully, combine two changes, write the new equation, and use the same four rules on the sine and cosine waves.

Why it works

A curve with no equation

Here is a curve labelled only y=f(x)y = \mathrm{f}(x). A question can print a graph like this and ask you to draw y=f(x)+3y = \mathrm{f}(x) + 3 or y=f(−x)y = \mathrm{f}(-x), with no formula to put in a table of values. What you have is the picture and one fact: f(x)\mathrm{f}(x) is the height of the curve above the number xx. Here f(0)=3\mathrm{f}(0) = 3, f(1)=0\mathrm{f}(1) = 0, f(2)=−1\mathrm{f}(2) = -1 and f(3)=0\mathrm{f}(3) = 0.-4-3-2-1123456-4-3-2-1123456(0, 3)(1, 0)(2, −1)(3, 0)y = f(x)xyEach change to the equation does one thing to these heights. Move a few key points, such as the lowest point and the crossings, then draw the same shape through them.

Keep reading — free

The rest of the explanation, plus 4 worked examples you step through move by move.

Start free

Takes a minute — no card.