Graph Transformations

Grade 7

The four transformations

Given a graph of y = f(x), you can produce new graphs by changing the equation. There are four key transformations to know:

y = f(x) + a: translate up by a (shift vertical). y = f(x + a): translate left by a (shift horizontal — the sign is the opposite of what you might expect). y = −f(x): reflect in the x-axis. y = f(−x): reflect in the y-axis.

The counter-intuitive horizontal shift

f(x + 3) shifts the graph left by 3, not right. Think of it as: the graph reaches the same output 3 units earlier. This catches students out every time — write it as "opposite to the sign inside the bracket".

Stretches

y = af(x): vertical stretch by scale factor a (from the x-axis). y = f(ax): horizontal stretch by scale factor 1/a (from the y-axis — again, the reciprocal surprises people).

Worked example

EXAMPLE

The graph of y = f(x) passes through (2, 5). Write down the coordinates of the image of this point under the transformation y = f(x − 3) + 1.

f(x − 3) shifts 3 right: x-coordinate becomes 2 + 3 = 5. Adding 1 shifts up: y-coordinate becomes 5 + 1 = 6. (5, 6).

Try this one

The diagram shows two curves. Curve A has equation y = f(x) and curve B is a transformation of curve A.

Curve B passes through (2, 1) and curve A passes through (2, 4) at the same x-value.

Express the equation of curve B in terms of f(x). [2]

Nova's hint:

This involves identifying a vertical stretch from the relationship between the y-values of two curves at the same x.

Ask yourself: 'If I know f(2) = 4 and the transformed curve gives 1 at x = 2, what scale factor multiplies f(x) to give that result?'

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