Generalized sine function

#[A generalized cosine function has the following form.][Una función coseno generalizada tiene la siguiente forma.]#

$f(x) = A\cos\left[\omega(x-\beta)\right] + C$

#[Thus,][Por lo tanto,]#

$A = {}$ #[amplitude][amplitud]#
$\omega = {}$ #[angular frequency][frecuencia angular]#
$\beta = {}$ #[phase shift (horizontal offset; the graph first reaches a maximum $\beta$ units to the right of the $y$-axis).][cambio de fase (desplazamiento horizontal; la gráfica primero alcanza un máximo $\beta$ 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 $A,\ \omega,$ and $C$ from the graph.

#[Graph of ][Gráfica de ]# $\bold{f(x) = A\cos\left[\omega(x-\beta)\right] + C}$

$\displaystyle A = \frac{\text{highest value} - \text{lowest value}}{2} \qquad \omega = \frac{2\pi}{P} \qquad \beta = \alpha + \frac{P}{4}$
#[$C = {}$ 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 $\beta$ (or $\alpha$) 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.

#[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 with a cosine function, we use the formulas above.

$\displaystyle A = \frac{\text{highest value} - \text{lowest value}}{2} \approx \frac{20 - 0}{2} = 10$ \\ $P = 12$ \\ $\displaystyle \omega = \frac{2\pi}{P} = \frac{2\pi}{12} = \frac{\pi}{6}$ \\ $\beta = {}$ #[Value of $t$ at first high point][Valor de $t$ en el primer punto alto]# ${}\approx 6$ \\ $C = {}$ #[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 $\displaystyle {} = A\cos\left[\omega(t-\beta)\right] + C$ \\ \t $\displaystyle {} =10\cos\left[\frac{\pi}{6}(t-6)\right] + 10$.

One for you:

Determine a generalized cosine function $f$ whose graph is shown.

$f(x) = {}$ BOX* ACTIVEMATH

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