### Home > A2C > Chapter 8 > Lesson 8.2.2 > Problem8-127

8-127.

Find an equation for each graph below.

1. Use the general equation $y = a · \sin\left(x − h\right) + k$.
Determine the value for each of the three parameters.

First identify a convenient locator point.
In this case, we will use:

$\left(\frac{\pi}{4}, 2\right)$

The $x-$value of the point represents the horizontal shift.
The $y-$value of the point represents the vertical shift.

$h=\frac{\pi}{4},k=2$

The amplitude (a) is the distance from the midline $\left(k\right)$ to the highest point.
In this case $k = 2$ and the highest point is $3$, so $a = 1$.
Since the graph increases from the locator point in the same way $y = \sin x$ does, a is positive.

1. See part (a).
Pay careful attention to the scale on the $y$-axis when determining $a$ and $k$.

1. If you choose $\left(\frac{\pi}{6},0\right)$ as your locator point, notice the first cycle of the graph is an inverted sine curve, so the a value will be negative.

See part (a).

$y=-\sin\left(x-\frac{\pi}{6}\right)+2\text{ or }y=\sin\left(x+\frac{5\pi}{6}\right) +2$

1. See parts (a) and (c).