Unit 5 · Topic 3.6

Sinusoidal Function Transformations

Translate, dilate, and reflect sine and cosine graphs while preserving the structure of one complete cycle.

Learning Goals

Interpret every parameter in factored sinusoidal form.
Determine amplitude, midline, range, period, and phase shift.
Recognize reflections and horizontal or vertical dilations.
Transform five parent-function landmarks into a new graph.
Construct and verify a sinusoidal equation from graph features.

1. Start with Factored Standard Form

\[y=A\sin(B(x-C))+D\quad\text{or}\quad y=A\cos(B(x-C))+D,\]

where \(A\ne0\) and \(B\ne0\). The factored input \(B(x-C)\) exposes the horizontal transformations correctly.

Parameter	Main effect
\(A\)	Amplitude and possible vertical reflection
\(B\)	Period and horizontal dilation
\(C\)	Phase shift
\(D\)	Midline and vertical shift

2. Translate Vertically with \(D\)

Adding \(D\) to the output moves every point vertically by \(D\) units:

\[y=f(x)+D.\]

The new midline is \(y=D\). A positive value shifts up; a negative value shifts down. Period and amplitude do not change.

3. Dilate and Reflect with \(A\)

The amplitude is the magnitude of the outside coefficient:

\[\text{amplitude}=|A|.\]

If \(|A|>1\), the graph stretches vertically; if \(0<|A|<1\), it compresses. If \(A<0\), outputs are reflected across the midline after scaling.

\[\text{range}=[D-|A|,D+|A|].\]

4. Compare Parent and Transformed Waves

The transformed wave is reflected and compressed horizontally around a shifted midline.

The reference wave and transformed wave differ in vertical placement, height, horizontal spacing, or orientation while retaining the same smooth repeating structure.

5. Change Period with \(B\)

Multiplying the input by \(B\) changes the horizontal scale by a factor of \(1/|B|\):

\[P=\frac{2\pi}{|B|}.\]

For \(|B|=3\), one cycle takes only \(2\pi/3\) input units. For \(|B|=1/4\), one cycle takes \(8\pi\) units. A larger \(|B|\) means a shorter period.

6. Shift Horizontally with \(C\)

In the factored input \(B(x-C)\), the phase shift is \(C\): right when \(C>0\), left when \(C<0\).

\[B(x-C)=0\quad\Longrightarrow\quad x=C.\]

This input corresponds to parent input 0. The inside sign appears opposite the direction of movement because the new input must compensate for the subtraction.

7. Factor before Reading the Shift

For

\[y=2\sin(4x+\pi)-3,\]

factor 4 from the complete input:

\[4x+\pi=4\left(x+\frac\pi4\right)=4\left(x-\left(-\frac\pi4\right)\right).\]

The phase shift is left \(\pi/4\), not left \(\pi\). The amplitude is 2, period is \(\pi/2\), and midline is \(y=-3\).

8. Transform the Five Key Inputs

For \(B>0\), start at \(x=C\) and advance by one quarter-period:

\[\Delta x=\frac{P}{4}=\frac{\pi}{2B}.\]

Parent input \(u\)	New input \(x=C+u/B\)	New output
0	\(C\)	\(D+A f(0)\)
\(\pi/2\)	\(C+P/4\)	\(D+A f(\pi/2)\)
\(\pi\)	\(C+P/2\)	\(D+A f(\pi)\)
\(3\pi/2\)	\(C+3P/4\)	\(D+A f(3\pi/2)\)
\(2\pi\)	\(C+P\)	\(D+A f(2\pi)\)

9. Analyze a Complete Cosine Transformation

Consider

\[y=-2\cos\left(3\left(x-\frac\pi6\right)\right)+4.\]

Amplitude: 2
Reflection: across the midline because \(A<0\)
Period: \(2\pi/3\)
Phase shift: right \(\pi/6\)
Midline: \(y=4\)
Range: \([2,6]\)

At \(x=\pi/6\), the cosine input is 0, so the reflected graph begins at its minimum \(y=2\).

10. Plot the Worked Example

The quarter-period is \((2\pi/3)/4=\pi/6\). Beginning at \(x=\pi/6\), the five outputs are

x	\(\pi/6\)	\(\pi/3\)	\(\pi/2\)	\(2\pi/3\)	\(5\pi/6\)
y	2	4	6	4	2

Connect these points smoothly and repeat every \(2\pi/3\).

11. Construct a Function from Characteristics

Build a sine function with amplitude 3, period 4, midline \(y=2\), and an upward midline crossing at \(x=1\).

\[|A|=3,\qquad B=\frac{2\pi}{4}=\frac\pi2,\qquad C=1,\qquad D=2.\]

\[y=3\sin\left(\frac\pi2(x-1)\right)+2.\]

The range is \([-1,5]\), and substituting \(x=1\) gives the midline value 2 with the parent sine orientation.

12. Choose Sine or Cosine Strategically

Convenient known landmark	Natural starting form
Upward midline crossing	Positive sine
Downward midline crossing	Negative sine
Maximum	Positive cosine
Minimum	Negative cosine

Different sine and cosine formulas can represent the same graph because the parent functions differ by a phase shift.

13. Track a Parent Point Algebraically

If \((u,f(u))\) lies on the parent graph and \(B\ne0\), then the corresponding transformed point is

\[\left(C+\frac{u}{B},\ D+A f(u)\right).\]

This rule handles horizontal and vertical changes at once. When \(B<0\), increasing parent inputs map in the opposite horizontal direction.

14. Verify with Technology

Predict midline, range, period, and phase shift first.
Choose a window that displays at least two cycles and the full range.
Trace the five calculated landmarks.
Check that consecutive matching extrema differ by one period.
Confirm the direction at the phase-shift input.

A graphing utility should verify the transformation analysis, not replace the parameter reasoning.

15. Common Errors

Reporting amplitude as \(A\) instead of \(|A|\).
Using \(2\pi/B\) without an absolute value for the period.
Reading phase shift before factoring the inside expression.
Moving right for \(x+C\) instead of left.
Ignoring the reflection created by \(A<0\).
Using period instead of quarter-period between adjacent key points.
Applying the vertical shift before scaling parent outputs in a point calculation.

Key Takeaways

Construct transformed sine and cosine functions from graph features and describe parameter effects.
Core relationship: \(y=A\cos(B(x-C))+D\)
Error check: Apply the horizontal shift from the factored form \(B(x-C)\).

Checkpoint · Topic 3.6

Analyze \(y=4\sin(2(x+\pi/3))-1\).
Rewrite \(y=-3\cos(6x-\pi)+5\) in factored form and state its phase shift.
List five key points for \(y=2\sin(\pi(x-2))+3\).
Construct a cosine function with maximum 7, minimum -1, period 10, and a maximum at \(x=4\).
Explain how a negative \(B\) changes the point-mapping direction.