Reflections & Stretches

Apply reflections and stretches to function graphs. Math 30-1 Alberta curriculum.

Lesson 1.7 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Vertical Stretches & Compressions. Learn how the coefficient 'a' in y = a·f(x) stretches or compresses graphs vertically, making them taller or flatter.
• Horizontal Stretches & Compressions. Discover how the coefficient 'b' in y = f(bx) affects the width of graphs—and why it works opposite to what you might expect!
• Combining Stretches & Compressions. See how vertical and horizontal scaling work together in y = a·f(bx) to transform graphs in both directions.
• Reflections. Understand how negative coefficients create reflections across the x-axis and y-axis, flipping graphs upside down or left-right.

Inside the lesson — a free preview

What is Vertical Scaling?

A vertical stretch or compression changes how tall or flat a graph looks without changing its width.

The general form is:

y = a · f(x)

where a controls whether the graph stretches or compresses vertically.

The Rule

Look at the value of |a| (the absolute value of a):

Stretch vs Compression Rules.

How It Works

The value a is a multiplier for all the y-values.

Point Transformation Rule. If (x, y) is on f(x), then (x, ay) is on y = a · f(x)

Example: Starting with f(x) = x². Original point: (2, 4)

For y = 2 · f(x): multiply the y-value by 2 → (2, 8)

For y = (1)/(2) · f(x): multiply the y-value by (1)/(2) → (2, 2)

Key Insight. The x-values stay the same. Only y-values change.

Invariant Points

An invariant point is a point on a graph which remains unchanged after a transformation is applied to it.

For Vertical Transformations. In y = a · f(x), a point (x, y) is invariant when:

y = ay

This occurs when y = 0

Therefore: All points on the x-axis are invariant for vertical stretches and compressions.

Example. For y = 3 · f(x):

The point (5, 0) is invariant → (5, 3 · 0) = (5, 0) ✓

The point (2, 4) is NOT invariant → (2, 3 · 4) = (2, 12) ✗

Vertical Stretch (|a| > 1)

When |a| > 1, the graph stretches vertically — it gets taller and steeper.

In the graph below, the black parabola represents the parent function, while the red parabola shows the vertically stretched version. Notice how the red graph is taller and steeper than the original.

Worked example: Quadratic

Example 1 — Quadratic. Parent function: f(x) = x^2

Transformed function: y = 3 · x^2

Analysis.

Points:

Point Transformation.

Result. The parabola becomes narrower and steeper.

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