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.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.