Generalized sine function
#[A
generalized sine function has the following form.][Una
función seno generalizada tiene la siguiente forma.]#
$f(x) = A\sin\left[\omega(x-\alpha)\right] + C$
#[Thus,][Por lo tanto,]#
- $A = {}$ #[amplitude][amplitud]#
- $\omega = {}$ #[angular frequency][frecuencia angular]#
- $\alpha = {}$ #[phase shift (horizontal offset; the graph first crosses the baseline $\alpha$ units to the right of the $y$-axis).][cambio de fase (desplazamiento horizontal; la gráfica primero cruza la línea base $\alpha$ unidades a la derecha del eje $y$).]#
- $C = {}$ #[vertical offset (the graph is moved $C$ units up).][desplazamiento vertical (la gráfica se mueve $C$ unidades hacia arriba).]#
The following graph shows what this curve looks like, with some formulas to get $A,\ \omega,\ \alpha,$ and $C$ from the graph.
#[Graph of ][Gráfica de ]# $\bold{f(x) = A\sin\left[\omega(x-\alpha)\right] + C}$
$\displaystyle A = \frac{\text{highest value} - \text{lowest value}}{2} \qquad \omega = \frac{2\pi}{P} \qquad \alpha = \beta - \frac{P}{4}$
#[$C = {}$ height of baseline: Average of highest and lowest values][$C = {}$ altura de la línea base: promedio de los valores alto y bajo]#
Note Increasing or decreasing $\alpha$ (or $\beta$) by the period $P$ or multiples of $P$ has no effect on the graph (and will be permitted in the interactive exercises) as we would be moving it horizontally that distance. For convenience we are using the lowest nonnegative value for $\alpha$ as shown in the graph.
#[Example][Ejemplo]#
The following graph shows the month-by-month average temperature in Germany over a period of two years.
To model the average temperature in Germany, we use the formulas above.
$\displaystyle A = \frac{\text{highest value} - \text{lowest value}}{2} \approx \frac{20 - 0}{2} = 10$
\\ $P = 12$
\\ $\displaystyle \omega = \frac{2\pi}{P} = \frac{2\pi}{12} = \frac{\pi}{6}$
\\ $\displaystyle \alpha = \beta - \frac{P}{4} \approx 6 - \frac{12}{4} = 3$
\\ $C = {}$ #[Average of highest and lowest values][promedio de los valores alto y bajo]# $\displaystyle \approx \frac{20+0}{2} = 10$.
#[Thus, our approximate model is][Por lo tanto, nuestro modelo aproximado es]#
$f(t)$ \t $\displaystyle {} = A\sin\left[\omega(t-\alpha)\right] + C$
\\ \t $\displaystyle {} =10\sin\left[\frac{\pi}{6}(t-3)\right] + 10$.