Video Transcript
Which of the following is the graph
of 𝑦 equals sin of two 𝑥?
We will consider each of the five
graphs given to determine which one has the same transformations as 𝑦 equals sin of
two 𝑥. First, we need to recall the
various transformations of the original 𝑦, or 𝑓 of 𝑥, equals sin of 𝑥
function. The first category of
transformations are called translations or shifts. A vertical shift up or down 𝑎
units is given by 𝑓 of 𝑥 plus 𝑎, whereas a horizontal shift left or right by
negative 𝑏 units is given by 𝑓 of 𝑥 plus 𝑏.
This kind of translation can be a
little challenging to get right, so we will review two quick examples. The notation 𝑓 of 𝑥 plus eight
means shift left eight, and the notation 𝑓 of 𝑥 minus eight means shift right
eight. The third type of transformation is
referred to as a vertical stretch or amplitude change. We recall the standard amplitude of
the sine function is one. So, if we multiply the function by
a scale factor 𝑐, we get a new amplitude of the absolute value of 𝑐. The last transformation we will
review is the horizontal stretch, also referred to as a period change. We recall the original sine
function has a period of 360 degrees. If we are given 𝑓 of 𝑑 times 𝑥,
this means the period is multiplied by a scale factor of one over 𝑑.
Now we will consider which
transformation is given by the function 𝑦 equals sin of two 𝑥. If we start with sin of 𝑥 equals
𝑓 of 𝑥, we should be able to determine what sin of two 𝑥 equals. By replacing 𝑥 with two 𝑥, we
determine that sin of two 𝑥 equals 𝑓 of two 𝑥, which we recognize as a period
change. In this case, 𝑑 equals two. So the scale factor applied to the
period is one over two. So, although the amplitude has not
changed, the original period of 360 is multiplied by one-half. Therefore, we are looking for the
sine graph that has a period of 180 degrees.
We will now consider option
(A). This appears to be the original
sine function, with an amplitude of one and a period of 360, which we have
highlighted in blue. We recognize the five familiar
coordinate points between zero and 360, where 𝑥 equals an angle measure in degrees
and 𝑦 equals sin of 𝑥. We will look for these five points
in our other sine graphs to determine if there has been a period change or other
transformation. We eliminate the first option
because it is a graph of the original sine function with no period change.
Moving on to option (B), we can
trace one period of sine, starting at the coordinate point 90, zero. This happens when we have a
𝑏-value of negative 90, which causes a horizontal shift of positive 90. We see all five points have been
shifted to the right 90 degrees. This horizontal shift preserves the
period length of 360 degrees. So this is not the function we are
looking for. So, we eliminate option (B) and
move on to option (C).
The first thing we notice about
option (C) is the period change. The period length has been
condensed, but we must determine if it’s been condensed to 180 degrees or not. If we highlight one period of this
sine curve, between negative 30 and 90 degrees, we come up with a period length of
120 degrees. So, we eliminate this option based
on the fact that the period is not the desired 180 degrees. It also appears this sine curve has
been shifted left by 30 degrees.
Moving on to option (D), we notice
another period change. We trace one period of this sine
curve, which happens to be 180 degrees exactly. This points to a period change with
a scale factor of one-half, which means 𝑑 equals two. This is the graph of 𝑓 of two 𝑥,
which is our 𝑦 equals sin of two 𝑥 function.
To make sure we’ve considered all
the options, we take a look at option (E). The first thing we notice is that
this sine curve is no longer centered around the 𝑥-axis. In this case, we have a vertical
shift of 𝑎 equals one. So, the period and amplitude stay
the same, but all points shift up one unit. Because of the vertical shift and
no period change in option (E), we can safely cross this possibility off our
list.
Option (D) is the only graph with
the correct amplitude of one and period of 180, without any horizontal or vertical
shifts. Therefore, (D) is the graph of 𝑦
equals sin of two 𝑥.
