Graph Transformations: Shifts

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.