Graph Transformations: Stretches

\( y=af(x) \;\text{(stretch scale }a\text{ in y)},\quad y=f(ax) \;\text{(scale }\tfrac{1}{a}\text{ in x)} \)
Graphs GCSE
Question 7 of 20
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\( \text{Apply }y=4f(x)\text{ to (2,1). \)

Hint (H)
Multiply y by 4.

Explanation

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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.

Worked examples

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  1. \( Apply y=2f(x) to the point (3,4). \)
    1. Vertical stretch factor 2.
    2. Multiply y by 2: (3,4) → (3,8).
    Answer: (3,8)
  2. \( Apply y=f(2x) to the point (6,5). \)
    1. Horizontal scale factor 1/2.
    2. Divide x by 2: (6,5) → (3,5).
    Answer: (3,5)
  3. \( Transform y=x^2 into y=4x^2. \)
    1. Multiply outside by 4.
    2. Stretch vertically by factor 4.
    Answer: \( y=4x^2 \)
  4. \( Transform y=√x into y=√(3x). \)
    1. Inside multiply by 3.
    2. Horizontal scale factor 1/3.
    3. Graph is compressed horizontally by 3.
    Answer: Horizontal compression by 3
  5. \( Apply y=0.5f(x) to the point (2,6). \)
    1. Vertical scale factor 0.5.
    2. Multiply y by 0.5: (2,6) → (2,3).
    Answer: (2,3)
  6. \( Transform y=sin(x) into y=sin(0.5x). \)
    1. Inside multiply by 0.5.
    2. \( Horizontal scale factor 1/0.5 = 2. \)
    3. Graph is stretched horizontally by 2.
    Answer: Stretched horizontally by 2
  7. \( Apply y=3f(x) to the point (-1,-2). \)
    1. Vertical scale factor 3.
    2. y multiplied by 3: (-1,-2) → (-1,-6).
    Answer: (-1,-6)
  8. \( Transform y=cos(x) into y=2cos(2x). \)
    1. Vertical stretch factor 2.
    2. Horizontal compression by 2.
    3. Amplitude doubles, period halves.
    Answer: \( Amplitude=2, period=π \)
  9. \( Apply y=f(0.25x) to (8,3). \)
    1. \( Horizontal scale factor 1/0.25=4. \)
    2. Divide x by 0.25: (8,3) → (32,3).
    Answer: (32,3)
  10. \( Transform y=x^3 into y=5x^3. \)
    1. Vertical scale factor 5.
    2. \( Equation is y=5x^3. \)
    Answer: \( y=5x^3 \)