Graph Transformations: Stretches

GCSE Graphs transformations stretches
Practice this formula All formulas
\( y=af(x) \;\text{(stretch scale }a\text{ in y)},\quad y=f(ax) \;\text{(scale }\tfrac{1}{a}\text{ in x)} \)

Statement

Stretches are transformations that change the size of a graph in one direction while keeping the other direction unchanged.

\[ y = a f(x) \quad \text{stretches the graph by scale factor } a \text{ in the y-direction} \]

\[ y = f(ax) \quad \text{stretches the graph by scale factor } \tfrac{1}{a} \text{ in the x-direction} \]

Why it’s true

  • In \(y = a f(x)\), each y-value is multiplied by \(a\). If \(a > 1\), the graph becomes taller (stretched vertically). If \(0 < a < 1\), the graph is squashed vertically.
  • In \(y = f(ax)\), each x-value is scaled by factor \(\tfrac{1}{a}\). If \(a > 1\), the graph is squashed horizontally. If \(0 < a < 1\), the graph is stretched horizontally.

Recipe (how to use it)

  1. Identify whether the stretch is vertical (\(a f(x)\)) or horizontal (\(f(ax)\)).
  2. For vertical: multiply all y-values by \(a\).
  3. For horizontal: divide all x-values by \(a\) (equivalent to scaling by \(1/a\)).
  4. Sketch key points after the transformation to redraw the curve.

Spotting it

  • Number outside function → vertical stretch/compression.
  • Number inside function (with \(x\)) → horizontal stretch/compression.

Common pairings

  • Reflections, when the scale factor is negative.
  • Translations, when combined with stretches for exam questions.

Mini examples

  1. Given: \(y=x^2\). Apply \(y=3x^2\). Result: Graph is stretched vertically by factor 3.
  2. Given: \(y=\sin(x)\). Apply \(y=\sin(2x)\). Result: Graph is squashed horizontally by factor 2 (period halves).

Pitfalls

  • Mixing inside vs outside: Outside affects y-values; inside affects x-values.
  • Forgetting reciprocal rule: \(f(ax)\) means a horizontal scale factor of \(1/a\).
  • Sign issues: Negative factors combine stretch with reflection.

Exam strategy

  • Write down whether the change is inside or outside before drawing.
  • Check the effect on one or two simple points.
  • For trigonometric graphs, remember that inside factors change the period (e.g., \(\sin(ax)\) has period \(\tfrac{2\pi}{a}\)).

Summary

Stretches alter the scale of a graph. Multiplying outside the function (\(a f(x)\)) changes the vertical scale. Multiplying inside the function (\(f(ax)\)) changes the horizontal scale by factor \(\tfrac{1}{a}\). Distinguishing inside from outside is the key to mastering graph stretches.