B: Algebra and functions: Understand the effect of simple transformations on the graph of y = f(x) including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax), and combinations of these transformations.

The idea

A transformation changes a graph's position, size or orientation. For y = af(x), every y-coordinate is multiplied by a, so the curve stretches vertically by factor a (a reflection in the x-axis if a < 0); points where f(x) = 0 stay fixed. For y = f(x) + a, the whole graph translates by a units parallel to the y-axis. For y = f(x + a), the graph translates by −a units parallel to the x-axis, so f(x + 2) moves left. For y = f(ax), x-coordinates are divided by a, giving a horizontal stretch of factor 1/a (a reflection in the y-axis if a < 0). Combinations apply these in turn, and sketching means marking key features such as intercepts, turning points and asymptotes after transformation.

What earns the marks

• Identifies y = af(x) as a vertical stretch of factor a, or a reflection in the x-axis when a is negative, leaving x-intercepts unchanged.
• States that y = f(x) + a translates the graph by a units parallel to the y-axis, moving every point including turning points and asymptotes.
• Recognises y = f(x + a) as a horizontal translation by −a units, so f(x + 3) shifts the graph three units left.
• Describes y = f(ax) as a horizontal stretch of factor 1/a, or a reflection in the y-axis when a is negative, leaving y-intercepts unchanged.
• Applies combinations in a consistent order, for example y = 2f(x − 1) as a translation right by 1 then a vertical stretch of factor 2.
• Sketches the transformed curve with correct key features: intercepts, stationary points, asymptotes and overall shape.

Common mistakes and how to put them right

• Treating f(x + a) as a shift right by a; correct this by testing a point, since f(x + 2) at x = 0 gives f(2), so the graph moves left.

• Applying a vertical stretch to x-coordinates or a horizontal stretch to y-coordinates.

  Put it right by remembering that changes inside the bracket affect x and changes outside affect y.

• Assuming a horizontal stretch of factor a rather than 1/a; correct by noting f(2x) halves each x-coordinate, so the stretch factor is 1/2.

Exam tips

• Sketch the original curve lightly first, then transform a few labelled key points to anchor the new graph.
• Use a numerical check such as comparing f(0) with the transformed value to confirm the direction of a translation.
• When several transformations are combined, state the order you are using and apply each one to the key points rather than redrawing freehand.