# How do you graph y = 2sin(4x) + 1, by using the period, domain, range and intercepts?

Dec 7, 2016

a) In the function $y = a \sin \left(b \left(x + c\right)\right) + d$, the period is given by $\frac{2 \pi}{b}$.

Call the period P.

$P = \frac{2 \pi}{4}$

$P = \frac{\pi}{2}$

$\therefore$The period is $\frac{\pi}{2}$ units.

b) In the function $y = a \sin \left(b \left(x + c\right)\right) + d$, the amplitude is $a$ and the vertical displacement is $d$.

The maximum is given by $a + d$.

The minimum is given by $a - d$.

Accordingly, the maximum is at $y = 3$ and the minimum is at $y = 1$. Therefore, the range is {y|1 ≤ y ≤ 3, y in RR}.

c) We have most of the information needed to graph from the above questions, but I recommend you find the intercepts (the y intercept and a general rule for the x-intercepts.

y intercept

$y = 2 \sin \left(4 \times 0\right) + 1$

$y = 2 \sin \left(0\right) + 1$

$y = 0 + 1$

$y = 1$

x intercepts

$0 = 2 \sin \left(4 x\right) + 1$

$- \frac{1}{2} = \sin \left(4 x\right)$

$\arcsin \left(- \frac{1}{2}\right) = 4 x$

$\frac{7 \pi}{6} , \frac{11 \pi}{6} = 4 x$

$x = \frac{7 \pi}{24} \mathmr{and} \frac{11 \pi}{24}$

Due to the periodicity of the sine function, $x = \frac{7 \pi}{24} + 2 \pi n$ and $\frac{11 \pi}{24} + 2 \pi n$ where $n$ is an integer.

We have all the information necessary to graph. My grapher won't let me label points, but you can look above and find a good many (using x-intercepts, periodicity, maximums and minimums).

Hopefully this helps!