Describing combined transformations, equations, ranges and inverses

Last time

Before we start: last time, in Composite and inverse functions, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Moving a graph around

Take any graph. Slide it, stretch it, flip it, and the equation changes in a precise way. Every Cambridge question on this is the same translation problem, run in one of two directions. Describe what happened from an equation, or write down the new equation from a description. The order of the steps is what examiners check.

The six moves

Adding outside moves the graph up. f of x plus a is a translation of 0, a. Adding inside moves it left. f of x plus a is a translation of minus a, 0. A number multiplying outside is a stretch parallel to the y axis. A times f of x has scale factor a. A number multiplying x inside is a stretch parallel to the x axis with scale factor one over a. A minus outside reflects in the x axis. A minus on the x reflects in the y axis.

Describe fully

Name the type of transformation, give the scale factor or vector, and the direction or line. For a stretch, state the axis it is parallel to. For a combination, also state the order. Without the order, you lose a mark. Use the word stretch for scale factors, translation with a vector, and reflection with a line.

A quadratic transformed

The curve y equals x squared becomes y equals 4 times x minus 3 squared, minus 8. A stretch comes with the 4, a translation with the 3 and the 8. Do the stretch first: scale factor 4 parallel to the y axis, giving y equals 4 x squared. Then translate by the vector 3, minus 8. Every point moves 3 right and 8 down. Check with one point: 1, 1 becomes 1, 4, then 4, minus 4. And 4 times 1 squared minus 8 is minus 4.