M151Q Precalculus · Unit 4: Trigonometric functions · on Final Exam
TRIG 7.1–7.3 Transformations, the general sinusoid, and solving equations
Study notes for Transformations, the general sinusoid, and solving equations in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- General sinusoid $y = A\sin(B(x - h)) + k$ (or cosine): amplitude $|A|$, period $2\pi/B$ (or $360^\circ/B$), horizontal shift $h$, midline $y = k$. Maximum $k + |A|$, minimum $k - |A|$.
- Model periodic data (tides, temperature, Ferris wheels): midline is the average of max and min, amplitude is half the difference, period from the cycle, shift from where the cycle starts.
- Graph in one period: mark five key points (start, quarter, half, three-quarter, end) using the period, then apply amplitude and midline.
- Solve $A\sin(Bx) + k = c$: isolate the sine, find the reference angle, list all solutions of the inner angle in the needed range, then divide by $B$. More solutions appear when $B > 1$.
Key formulas
General sinusoid
$$y = A\sin\big(B(x - h)\big) + k,\quad \text{amp } |A|,\ \text{period } \frac{2\pi}{B},\ \text{midline } y = k$$
From max and min
$$k = \frac{\max + \min}{2},\qquad |A| = \frac{\max - \min}{2}$$
Worked example
Water depth at a dock varies from 2 m to 8 m with a 12-hour cycle, high tide at $t = 3$. Write $d(t)$ with cosine.
Show the solution
Midline $5$, amplitude $3$, period 12 so $B = 2\pi/12 = \pi/6$, cosine peaks at the shift: $d(t) = 3\cos\!\left(\dfrac{\pi}{6}(t - 3)\right) + 5$.
Common mistakes
- Using B as the period instead of 2π/B.
- Applying the horizontal shift before factoring out B: sin(2x − π) has shift π/2, not π.
- Forgetting the extra solutions when the period is shortened.
Exam tipWrite the four numbers (amplitude, period, shift, midline) before the formula. The formula is just bookkeeping.
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