Function Transformations UPCAT Reviewer: Shifts, Reflections, and Stretches

TEACHER ABI UPCAT MATHEMATICS

Function Transformations

Changes inside the function move inputs horizontally; changes outside alter outputs vertically.

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Function Transformations

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Transformations move or reshape a parent graph predictably

Start with a parent function y = f(x). In the transformed rule below, h controls a horizontal shift, k controls a vertical shift, and a changes the outputs.

y = a f(x−h) + k

Inside changes work in the opposite visible direction: x−h shifts right h, while x + h shifts left h. Outside changes follow their signs.

The vertex makes each transformation easy to track.

Inside affects x

f(x−h) shifts right h; f(x + h) shifts left h. Horizontal scaling also uses the reciprocal factor.

Outside affects y

af(x) + k multiplies every output by a, then shifts all outputs by k. A negative a reflects across the x-axis.

DO IT FAST

Track one landmark point before imagining the whole graph

Choose the vertex, intercept, or any known point (a,b). Under y = cf(x−h) + k, it maps to (a + h, cb + k). This often answers the question without sketching the complete graph.

Why it works

The transformed rule changes either the input supplied to f or the output produced by f. Point mapping makes those two effects explicit.

WORKED EXAMPLES

Five forms you should recognize

1. Horizontal shift

Problem: Compare g(x) = f(x−4) with f(x).

f(x−4) → right 4

Why: The new graph reaches the old input a when x−4 = a, or x = a + 4.

2. Vertical shift

Problem: Compare g(x) = f(x) + 6 with f(x).

(a,b) → (a,b + 6)

Why: Six is added to every output while inputs stay unchanged.

3. Reflection

Problem: Reflect y = f(x) across the x-axis.

y = −f(x); (a,b) → (a,−b)

Why: Negating outputs reverses their vertical direction.

4. Combined transformation

Problem: Interpret y = −2f(x + 1) + 4.

left 1 → vertical stretch 2 → reflect across x-axis → up 4

Why: The input changes first; outside operations then transform the output.

5. Point mapping

Problem: (3,5) lies on f. Find its image on y = f(x−2) + 1.

(3,5) → (5,6)

Why: The entire graph moves right 2 and up 1.

COMMON TRAPS

Check before you commit

• Reading x−h as a shift left instead of right
• Confusing −f(x) with f(−x)
• Multiplying x-coordinates by an outside vertical scale factor
• Using the horizontal multiplier directly instead of its reciprocal
• Moving a vertex but forgetting a reflection or stretch
• Trying to sketch every point when one landmark point is enough

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Core Practice

Use mixed forms with less scaffolding.

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Apply the competency in unfamiliar representations.

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QUICK ANSWERS

Function Transformations FAQ

Why do horizontal shifts seem backward?

Because f(x−h) receives the old input a when the new x-value is a + h, so points move right.

What is the difference between −f(x) and f(−x)?

The first reflects outputs across the x-axis; the second reflects inputs across the y-axis.

How do I map a known point?

For y = cf(x−h) + k, map (a,b) to (a + h,cb + k).

RELATED COMPETENCIES

Continue your mathematics review.

Function EvaluationDomain and RangeQuadratic Graph FeaturesPiecewise Functions

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