Sinusoidal Function Model Builder

The window runs from about −1 to 13 horizontally and −7 to 7 vertically. The wave is coral. A dashed teal line marks the midline $y = d$, two lighter dashed lines mark the maximum and minimum $d \pm |a|$, and a white dot marks the point $(c, d)$, where this sine wave crosses its midline going up.

The panel writes the equation with b rounded to two decimals, for example y = 3·sin(0.79·(x − 1.50)) + 1, then lists the amplitude, the period with its b value, the phase shift with "right" or "left", and the midline. Because b is rounded, use the period, not the printed b, when you need an exact value such as $\frac{\pi}{4}$.

For $y = a\sin\big(b(x - c)\big) + d$ with $b > 0$: $$\text{amplitude} = |a|, \qquad \text{period } P = \frac{2\pi}{b} \ \Longleftrightarrow\ b = \frac{2\pi}{P}, \qquad \text{midline } y = d.$$

The phase shift is c units to the right: the graph of $\sin x$ is translated so that its rising midline crossing moves from $x = 0$ to $x = c$. The AP Precalculus course description writes the general form as $a\sin\big(b(\theta + c)\big) + d$; with a plus sign inside, the shift is $c$ units to the left. Either way, factor b out first and read the shift from what is subtracted from x.

From features to an equation:

• $d = \dfrac{\max + \min}{2}$ and $|a| = \dfrac{\max - \min}{2}$.
• P is the horizontal distance between two consecutive maxima (or twice the distance from a maximum to the next minimum).
• For a sine model, c is an x-value where the graph crosses the midline going up. A maximum occurs a quarter period later, at $c + \frac{P}{4}$.

The same wave can be written with cosine, starting at a maximum instead: $a\cos\big(b(x - x_{\max})\big) + d$.

Step 1: midline and amplitude. $d = \frac{4 + (-2)}{2} = 1$ and $a = \frac{4 - (-2)}{2} = 3$.

Step 2: period. Maximum to the next minimum is half a period: $7.5 - 3.5 = 4$, so $P = 8$ and $b = \frac{2\pi}{8} = \frac{\pi}{4} \approx 0.785$.

Step 3: phase shift. The rising midline crossing comes a quarter period (2 units) before the maximum: $c = 3.5 - 2 = 1.5$.

Model: $$y = 3\sin\!\Big(\frac{\pi}{4}(x - 1.5)\Big) + 1, \qquad \text{or equivalently}\qquad y = 3\cos\!\Big(\frac{\pi}{4}(x - 3.5)\Big) + 1.$$

Check a value: at $x = 5$, $y = 3\sin\big(\frac{\pi}{4}\cdot 3.5\big) + 1 = 3\sin(2.749) + 1 \approx 3(0.383) + 1 \approx 2.15$, which sits between the midline and the maximum, as it should just after the peak.

Check it in the simulator: set a = 3, P = 8, c = 1.5, d = 1. The equation reads y = 3·sin(0.79·(x − 1.50)) + 1, the dot sits at (1.5, 1), the peak touches the upper dashed line y = 4 at x = 3.5 and the trough touches y = −2 at x = 7.5.

Sinusoidal modeling (Topics 3.6 and 3.7) is a favorite for free-response questions built on a context such as tides, temperatures or a Ferris wheel. You may be given a table of values or a graph and asked to find a, b, c and d, then use the model to estimate a value or identify where the function is increasing, decreasing, concave up or concave down.