Graph Transformations: Shifts

\( y=f(x)+a \;\text{(up by a)},\quad y=f(x-a) \;\text{(right by a)} \)
Graphs GCSE
Question 9 of 20
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\( \text{Shift the point }(7,-1)\text{ using }y=f(x+2). \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Shift left by 2.

Explanation

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Statement

Shifts, also called translations, move a graph without changing its shape. The graph slides either vertically or horizontally by a fixed amount.

\[ y = f(x) + a \quad \text{shifts the graph up by } a \]

\[ y = f(x - a) \quad \text{shifts the graph right by } a \]

Why it’s true

  • Adding \(+a\) outside the function changes the y-values directly. Every point moves up (if \(a > 0\)) or down (if \(a < 0\)).
  • Replacing \(x\) with \(x-a\) shifts the inputs. To get the same output as before, you must move to the right by \(a\). If \(a > 0\), the graph shifts right; if \(a < 0\), it shifts left.

Recipe (how to use it)

  1. Start with the base graph \(y = f(x)\).
  2. For vertical shifts: add \(a\) outside the function → \(f(x) + a\).
  3. For horizontal shifts: replace \(x\) with \(x - a\).
  4. Sketch key points, then slide the whole graph by the correct distance.

Spotting it

  • \(+a\) outside → vertical movement.
  • \(-a\) inside → horizontal movement.

Common pairings

  • Reflections (flipping the graph).
  • Stretches and compressions.

Mini examples

  1. Given: \(y = f(x) = x^2\). Apply: \(y = x^2 + 3\). Result: Graph shifts up by 3.
  2. Given: \(y = f(x) = |x|\). Apply: \(y = |x-2|\). Result: Graph shifts right by 2.

Pitfalls

  • Confusing signs: \(f(x-3)\) means right 3, not left 3.
  • Mixing inside vs outside: Inside affects x (horizontal), outside affects y (vertical).
  • Forgetting negatives: \(f(x+2)\) means shift left by 2.

Exam strategy

  • Label shifts clearly before drawing.
  • Test a simple point like (0,0) to confirm the direction.
  • Remember: outside = up/down, inside = left/right.

Summary

Shifts slide a graph without changing its appearance. Vertical shifts come from adding outside the function (\(f(x)+a\)). Horizontal shifts come from adjusting the input (\(f(x-a)\)). Keeping track of the sign is the key to success.

Worked examples

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  1. \( Shift the point (2,3) using y=f(x)+4. \)
    1. Add 4 to the y-value.
    2. (2,3) → (2,7).
    Answer: (2,7)
  2. \( Shift the point (5,-1) using y=f(x-3). \)
    1. Subtract 3 from the x-coordinate.
    2. (5,-1) → (8,-1).
    Answer: (8,-1)
  3. \( Graph of y=x^2 shifted up by 2. \)
    1. \( Start with y=x^2. \)
    2. \( Add 2 outside: y=x^2+2. \)
    Answer: \( y=x^2+2 \)
  4. \( Graph of y=√x shifted right by 5. \)
    1. Replace x with x-5.
    2. \( Equation: y=√(x-5). \)
    Answer: \( y=\sqrt{x-5} \)
  5. \( Point (0,0) shifted by y=f(x)+6. \)
    1. Increase y by 6.
    2. (0,0) → (0,6).
    Answer: (0,6)
  6. \( Graph of y=|x| shifted left by 4. \)
    1. Replace x with x+4.
    2. \( Equation: y=|x+4|. \)
    Answer: \( y=|x+4| \)
  7. \( Graph of y=x^3 shifted down by 7. \)
    1. \( Equation: y=x^3. \)
    2. \( Shift down by 7: y=x^3-7. \)
    Answer: \( y=x^3-7 \)
  8. \( Shift the point (-3,2) using y=f(x-1). \)
    1. x increases by 1.
    2. (-3,2) → (-2,2).
    Answer: (-2,2)
  9. \( Graph of y=1/x shifted right by 2. \)
    1. Replace x with x-2.
    2. \( Equation: y=1/(x-2). \)
    Answer: \( y=\tfrac{1}{x-2} \)
  10. \( Shift the point (4,-5) using y=f(x)+3. \)
    1. Add 3 to y-coordinate.
    2. (4,-5) → (4,-2).
    Answer: (4,-2)