We want to determine an equation
of the given graph using the sine function.
We will determine an equation in the form
y equals a times sine of the quantity bx plus c plus d
where the absolute value of a is equal
to the amplitude, two pi divided by b is the period,
negative c divided by b is the phase shift
or horizontal shift, and d is the vertical shift.
The first step is to highlight one period
of the given sine function.
Let's begin by focusing on the interval
from negative four to positive six.
Notice over this interval, we do have a complete graph
of one period of the given sine function.
If you're wondering why I focused on this part,
notice how by focusing on this period,
there is no reflection across the midline
because the pattern is midline, maximum,
midline, minimum, midline
which is the same as the basic sine function shown here.
If we focused on a different interval,
we would get a different equation,
but it would be equivalent.
The first thing I notice here
probably is that the midline
is the x-axis, and therefore, the midline
is y equals zero, and there is no vertical shift.
And therefore, d is equal to zero.
Next, remember the amplitude is the distance
from the midline to a maximum
or the distance from the midline to a minimum.
And notice how this distance here is three units,
and therefore, the absolute value of a must be three.
And again, because of the pattern of midline
to maximum, midline to minimum, midline,
there is no reflection across the midline,
and therefore, a is equal to positive three.
And now, using the equation two pi divided by b,
we can determine the value of b
because we know the period is 10 units.
We know that two pi divided by b must equal 10.
Now we need to solve for b.
Let's multiply both sides by b.
b divided by b simplifies to one here,
giving us the equation two pi is equal to 10 b.
Dividing both sides by 10, we have b
is equal to two pi divided by 10
or 2/10 pi which simplifies to 1/5 pi or pi over five.
So now we know b is equal to pi over five.
And now let's determine c by identifying
the phase shift or horizontal shift.
We normally start the basic sine function
at the origin, but notice here, the graph is shifted left
four units, and therefore, the phase shift
or horizontal shift is negative four.
And therefore, negative c divided by b
must equal negative four.
So negative c divided by b must equal negative four.
And we know b is pi over five
which gives us negative c divided by pi over five
must equal negative four.
And now, let's clear the fraction from the equation
by multiplying both sides by pi over five.
Simplifying pi over five divided by itself
simplifies to one, giving us negative c
equals negative four pi over five.
And therefore, c is equal to positive four pi over five.
Now we have all the information we need
to write a possible equation for the given graph.
One possible equation is y equals,
again, a times sine of the quantity
bx plus c plus d will give us three
times the sine of bx plus c is pi over five x
plus four pi over five.
And because d is zero, we don't need plus zero.
This is one possible equation for the given graph.
Let's see if we can find another possible equation.
This time, where a is negative,
we're using a reflection across the midline.
So this time, let's focus on the interval
starting at positive one and moving to the right.
Notice how we don't have a complete graph
of one period of the sine function moving
in this direction, but as long as we can determine
the period which we already know,
we can still use this piece of the graph
to determine a second equation
that would give us the same graph.
Again, we still know there's no vertical shift,
and therefore, d is equal to zero.
The amplitude is still positive three.
But again, looking at the pattern of the graph,
because we now have a pattern
of midline, minimum, midline, maximum, midline,
we have a reflection across the midline.
And therefore, a is not positive three,
a is negative three.
We also know the period is still 10 units,
and therefore, b is still going to be pi over five.
But because the horizontal shift or phase shift
is different, c is going to be different.
Notice using this piece of the graph,
the phase shift is right one unit, and therefore,
negative c divided by b must equal positive one.
So using the equation negative c divided by b
equals positive one, we now need to find
a new value for c.
And again, we know b is pi over five
which gives us negative c
divided by pi over five equals one.
Again, we multiply both sides by pi over five.
Simplifying, we get negative c is equal
to positive pi over five, and therefore,
positive c is equal to negative pi over five.
So now we know c must be negative pi over five
if we're using this piece of the graph
to determine the equation.
So now we can come up with a different equation
for the same graph.
We could have also used the equation
y equals negative three times sine
of the quantity bx plus c which gives us
pi over five x, and then, plus c is
plus negative pi over five or just minus pi over five.
So both of these equations are correct
and will produce the given graph.
I hope you found this helpful.