**Exploration of Sine Curves**

**
**by

Chad Crumley

This exploration is of the function ** y =
a sin b(x - h) + k**, where a, b,
h, and k are different values.

In particular, how do these values transform the graph of ** y= sin x**. Before we begin, here is what that graphs look like
with a, b equal to 1 and h, k equal to 0.

**The Constant ***a*

*
*As

Thus *a* appears to stretch the graph.
The graph is stretched vertically (both in the positive and negative direction)
to the value of *a*. For example,
the maximum and minimum values reachedby the blue graph (when *a* = 5) is 5. Starting at the origin, the positive
value of *a* forces the graph to go
up first.

What happens as *a* is positive and
decreasing? (** a** = 1 is in red,

The same results from above except that the value of *a* is decreasing and the graph doesn't stretch outward
but is squeezed inward.

When *a* is negative? (** a** = 1 is in red,

Notice the graphs are reflections of the parent graph *y = sin x* over the x-axis and 'stretched' according the value
of *a*. Also, the x-intercepts (or
zeros) are the same for the graphs. The zeros are n*(pi) where n is any
integer. Starting at the origin, the negative value of *a* forces the graph to go down first.

The absolute value of *a* is called the **amplitude**
of the graph.

**The Constants ***h, k
*

For this exploration,

As

As *h* decreases? (** h** = 0 in red,

Conclusions: The graphs appear to translate right with a larger *h* and translate left with a smaller *h*.

Now let *h* = 0 and let's
investigate *k*.

Here are graphs with a =1, *b* =1,
and *h* = 0. ** k** = 0 is in red,

Conclusions: The graphs appear to translate up with a positive value for *k* and down with a negative value of *k*. Along with the transition, the y-intercepts for the
curves are the *k* values.

**The Constant ***b
*

For this exploration,

As

Comparing the graphs to the parent graph (red), the graphs looked compressed
(as if a force was pushing in on a spring). Also, the purple graph completes 2
complete cycles from 0 to 2 pi and the blue graph completes 3 cycles. Comparing
the period of the graphs, the period of the red graph is 2 pi, the period of
the purple graphs is pi, and the period of the blue graph is 2 pi / 3.

A *b* decreases? (** b** = 1 in red,

As *b* becomes smaller, the graphs are
pulled outward (like an over stretched spring). The purple graph completes one
cycle from 0 to 12 pi, which is twice as long as the parent. The blue graph
completes one cycle from 0 to 8 pi, which is 4 times as long as the parent
graph.

*b* remains positive or the graph will
reflect across the x-axis because of the opposites theorem: *sin (-x)
= -sin (x)*.

**Summary
**

The absolute value of

The period represents 2 pi / (the absolute value of