Transformations

Apply transformations to sinusoidal functions. Math 30-1 Alberta Grade 12.

Lesson 4.7 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Stretches & Compressions. Understand how the parameters a and b in y = a·sin(bx) create vertical and horizontal stretches and compressions, changing the amplitude and period of the graph.
• Horizontal Phase Shift. Learn how the parameter c in y = a sin(b(x - c)) + d creates horizontal translations (phase shifts) of sinusoidal graphs.
• Vertical Shift. Explore how the parameter d in y = a sin(b(x - c)) + d moves the graph up or down and changes the midline, range, and center of oscillation.
• Full Sinusoidal Transformations. Master the complete transformation equation y = a sin(b(x - c)) + d by combining all parameters to analyze amplitude, period, phase shift, and vertical shift together.

Inside the lesson — a free preview

Period & Amplitude — What Do They Mean as Transformations?

We've already seen that period is how long a trig cycle takes, and amplitude is how tall it gets. But these aren't just measurements — they're the result of stretching and compressing the graph. Let's unpack exactly what's happening geometrically.

Starting Point: The Parent Function

The Parent Function. Every transformed trig function is built from a parent function. For sine, that's:

y = (x)

• Amplitude = 1

• Period = 2π

• No stretch, no compression — the "default" shape.

When we write a transformed version, the two key numbers a and b control the stretches:

y = a · (bx)

• a → controls vertical stretch (amplitude)

• b → controls horizontal stretch (period)

Vertical Stretches & Compressions (the a value)

The a Value — Vertical Effect. Multiplying the outside of a function by a stretches or compresses it vertically — away from or toward the x-axis.

• If |a| > 1: the graph is vertically stretched — it gets taller. Amplitude increases.

• If 0 < |a| < 1: the graph is vertically compressed — it gets shorter. Amplitude decreases.

• If a is negative: the graph is also reflected over the x-axis.

The amplitude of y = a(bx) is simply |a|. The period is unchanged.

Horizontal Stretches & Compressions (the b value)

The b Value — Horizontal Effect. Multiplying inside the function — on the x — stretches or compresses it horizontally. This is counterintuitive: the effect is opposite to what you'd expect.

• If b > 1: the graph is horizontally compressed — cycles happen faster. Period decreases.

• If 0 < b < 1: the graph is horizontally stretched — cycles happen slower. Period increases.

Period (radians) = (2π)/(b) Period (degrees) = (360°)/(b)

The bigger b is, the more cycles fit in the same space → shorter period. The smaller b is, the fewer cycles → longer period.

Why is the horizontal effect opposite to what you'd expect?

When b = 2, you're asking the function to reach its full cycle in half the x-distance. The x-values are effectively being "divided by 2," so the wave gets squished. When b = (1)/(2), each x-step counts for less, so the wave stretches out.

Worked example: Vertical Stretch: y = 3(x)

Example 1 — Vertical Stretch: y = 3(x). Step 1 — Identify a and b.

a = 3, b = 1

Step 2 — Find the amplitude.

Amplitude = |a| = |3| = 3

The graph reaches up to y = 3 and down to y = -3.

Step 3 — Find the period.

Period = (2π)/(b) = (2π)/(1) = 2π

Unchanged from the parent.

→ This is a vertical stretch by a factor of 3.

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