Trigonometry Section 2.6 – Phase Shift and Composite Transformations

Administrative Announcements

• Upcoming Test (Test #1)
  • Opens Thursday, Feb 18 at 10 PM.
  • Covers $\text{Sections }2.1\;\text{to}\;2.6$ (entire Chapter 2).
  • Review packet will be posted this afternoon; instructor urges students to work through it 3–5 times for a “friendly” test experience.
  • Full credit on the test requires complete written processes. Answers without justification (e.g.
    straight from a graphing calculator) earn 0 pts.
• Homework
  • Section 2.6 homework due next Thursday.
  • Extra practice set (“D-12”) with answers (and rough graphs) will be posted tonight.
• Next Class
  • Tuesday session finishes Section 2.5 (considered “friendlier”).
  • Chapter 3 begins after the test.
• Miscellaneous
  • Severe cold & possible snow Sunday/Monday—stay inside and study.
  • Light-hearted reminder: “If you go to the movies or Burger King, take the review with you.”

Correction From Previous Lecture (Amplitude Placement)

• Last time an exercise with amplitude $0.75$ was graphed incorrectly.
• Error: Instructor mistakenly added/subtracted the amplitude to $\pm1$, labeling extrema at $\pm1.75$.
• Correct rule: Graph oscillates between $+0.75$ and $-0.75$ (no shift to $\pm1$).
• Reinforced principle:
  • Amplitude modifies vertical stretch only, not the vertical center unless an explicit vertical shift is present.

Transformation Review

• A general sine or cosine model:
$y = A\,\sin\big(B(x-C)\big) + D \quad\text{or}\quad y = A\,\cos\big(B(x-C)\big)+D$

• Amplitude $|A|$
  • Graph ranges between $D\pm|A|$.
  • A negative $A$ causes a vertical reflection (flips the curve across the midline).
• Period $P$
  • $P = \dfrac{2\pi}{|B|}$.
  • To update the key $x$-marks $\bigl(0,\tfrac{\pi}{2},\pi,\tfrac{3\pi}{2},2\pi\bigr)$ divide each by $|B|$.
• Phase Shift $C$
  • Horizontal translation determined from $B(x-C)=0 \Rightarrow x=C$.
  • Graph starts at $x=C$ (for sine) or at maximum/minimum accordingly (for cosine).
• Vertical Shift $D$
  • Translates graph up (D>0) or down (D<0).

Concept of Phase Shift (Section 2.6 Focus)

• Phase shift moves the entire curve left/right without affecting amplitude, period, or midline.
• Found by solving $B(x-C)=0$ or by factoring $B$:
  $B\bigl(x-C\bigr)=0 \;\longrightarrow\; x=C.$
• Two common strategies:
  1. Equation method: Set the inside to zero and solve for $x$.
  2. Factoring method: Factor $B$ then read the shift directly.
• Important: Do not treat the phase-shift sign as the even/odd test for bringing a minus sign outside; sign governs horizontal position only.

Detailed Worked Examples

Example 1

Given $y = 3\sin\bigl(2x-\pi\bigr)$

• Amplitude: $A=\pm3$
  • Range $[-3,3]$.
• Period: $P=\dfrac{2\pi}{2}=\pi$.
  • Original 5 key marks $\bigl[0,\tfrac{\pi}{2},\pi,\tfrac{3\pi}{2},2\pi\bigr]$ divide by $2$ → $0,\tfrac{\pi}{4},\tfrac{\pi}{2},\tfrac{3\pi}{4},\pi$.
• Phase shift:
  $2x-\pi=0\;\Rightarrow\;x=\tfrac{\pi}{2}.$
  • Graph starts at $x=\tfrac{\pi}{2}$.
• No vertical shift; no reflection (positive $A$).
• Five plotted points (shifted right) produce a sine curve beginning at the midline, rising to $3$, etc.

Example 2

$y=-2\cos\bigl(4x+3\pi\bigr)$

• Amplitude: $\pm2$; negative → reflects vertically.
• Period: $\dfrac{2\pi}{4}=\dfrac{\pi}{2}$. Points: $0,\tfrac{\pi}{8},\tfrac{\pi}{4},\tfrac{3\pi}{8},\tfrac{\pi}{2}$.
• Phase shift: $4x+3\pi=0\Rightarrow x=-\dfrac{3\pi}{4}$ (left shift).
• Baseline midline $y=0$ (no $D$).
• Reflected cosine therefore starts at a minimum of $-2$ at $x=-\dfrac{3\pi}{4}$, peaks at $+2$ one-quarter period later, etc.

Example 3

$y=-2\cos\bigl(-2x+\tfrac{\pi}{2}\bigr)+2$

• Amplitude: $2$ with reflection.
• Period: $\dfrac{2\pi}{|-2|}=\pi$.
• Phase shift (solve or factor):
  $-2x+\tfrac{\pi}{2}=0\Rightarrow x=\tfrac{\pi}{4}$ (right shift).
• Vertical shift: $D=+2$ → entire curve raised 2 units.
• Procedure instructor used:
  1. Plot raw reflected cosine between $\pm2$.
  2. Translate every point up 2 → final range $[0,4]$.

Example 4

$y=-\tfrac12\sin\bigl(4x-\tfrac{5\pi}{4}\bigr)-1$

• Amplitude: $\frac12$ (vertical reflection).
• Period: $\frac{2\pi}{4}=\frac{\pi}{2}$.
• Key-mark list (after dividing by 4): $0,\tfrac{\pi}{8},\tfrac{\pi}{4},\tfrac{3\pi}{8},\tfrac{\pi}{2}$.
• Phase shift: $4x-\tfrac{5\pi}{4}=0\Rightarrow x=\tfrac{5\pi}{16}$ (right shift ≈0.313 rad).
• Vertical shift: $D=-1$ (move entire sine curve down 1). Final midline $y=-1$, extreme values $-1\pm\tfrac12$.
• Instructor demonstrated plotting five points starting at $x=\tfrac{5\pi}{16}$ and then translating downward.

Example 5 – Building an Equation From Specs

“Write the cosine function with amplitude 2, period $\pi$, phase shift $\tfrac{\pi}{2}$.”

• Start: $y = 2\cos\bigl(B(x-C)\bigr)$.
• Find $B$:
  $P=\pi=\frac{2\pi}{|B|}\;\Rightarrow\;B=2.$
• Insert phase shift: $y = 2\cos\bigl(2(x-\tfrac{\pi}{2})\bigr)$.
• Distribute to match instructor’s final form:
  $y=2\cos\bigl(2x-\pi\bigr).$

Example 6 – Amplitude 3, Period $\tfrac{\pi}{4}$, Phase Shift $-\tfrac{5\pi}{6}$

• $B=\dfrac{2\pi}{P}=\dfrac{2\pi}{\tfrac{\pi}{4}}=8.$
• $y=3\cos\bigl(8(x+\tfrac{5\pi}{6})\bigr)$ (plus sign because shift is left). Instructor distributed:
  $y=3\cos\bigl(8x+\tfrac{40\pi}{6}\bigr)=3\cos\bigl(8x+\tfrac{20\pi}{3}\bigr).$

Example 7 – Negative Fractional Amplitude

Given specs: amplitude $-\tfrac{3}{8}$, period $\tfrac16$, phase shift $-\tfrac{3\pi}{4}$.

• $B=\dfrac{2\pi}{P}=2\pi\cdot6=12\pi.$
• Model:
  $y=-\tfrac{3}{8}\cos\Bigl(12\pi \bigl(x+\tfrac{3\pi}{4}\bigr)\Bigr)$ → instructor distributed to obtain
  $y=-\tfrac{3}{8}\cos\bigl(12\pi x+9\pi\bigr).$

Handling “π in the Denominator” (Graph-Starting Decimal Trick)

• When phase-shift algebra places $\pi$ in the denominator, convert to decimal to locate the first point quickly.
  Example: $3\pi x-\tfrac56=0 \Rightarrow x=\dfrac{5}{18\pi}\approx0.088.$
  • Since 0<0.088<0.1 it lies very close to the origin; plot accordingly.

Graphing Protocol & Common Mistakes Highlighted

• Graph must show:
  1. Midline.
  2. Five key points over one period.
  3. Correct extrema labels (after amplitude/vertical shifts).
  4. Proper direction (reflection awareness).
• Frequent student errors:
  • Adding amplitude to $\pm1$ instead of centering around midline.
  • Treating inner negative as odd/even property and moving it out front (incorrect for phase shift).
  • Forgetting to divide all five canonical $x$ points by $B$.
  • Applying vertical shift before plotting the base curve.

Instructor’s Pedagogical Metaphors & Advice

• “Review is your boyfriend/girlfriend—take it everywhere, even to the bathroom.”
• When the class is quiet, instructor jokes about being “scared like when my wife is quiet.”
• Encourages mistakes during practice: “Your mistakes are beautiful here; mine are worse.”

Ethical / Practical Implications

• Emphasis on honesty in showing work; calculators allowed for checking but not as sole solution source.
• Responsibility to practice repeatedly to internalize transformations rather than memorize isolated answers.

Real-World / Previous-Lecture Connections

• All transformation rules parallel those used in Section 2.4 (vertical stretch, reflection, vertical shift); Phase shift in 2.6 simply completes the toolkit.
• Building vs. reading equations: Inverse skills valuable for modeling periodic real-life data (e.g.
  tides, sound waves, alternating currents).

Numerical & Formula Summary

• Period formula: $P=\dfrac{2\pi}{|B|}$.
• Phase shift: $C=\dfrac{\text{constant}}{B}$ from $B(x-C)=0$.
• Key $x$ marks after period change: \bigl{0,\tfrac{P}{4},\tfrac{P}{2},\tfrac{3P}{4},P\bigr}.
• Vertical reflection if A<0; horizontal reflection never occurs in these forms (would require negative $B$ with even/odd analysis).

Study Checklist Before Test #1

• [ ] Know how to compute amplitude, period, phase shift, vertical shift from any $y = A\,f\bigl(B(x-C)\bigr)+D$.
• [ ] Practice both graphing from equation and writing equation from specs.
• [ ] Re-work every review question until processes feel “muscle-memory” level.
• [ ] Show every algebraic step on paper; annotate key points on sketches.
• [ ] Use the D-12 practice set for extra graphing drills.