Video Transcript
The figure shows the graph of 𝑦
equals sin of 𝑥. Which of the following is a graph
of 𝑦 equals sin of 𝑥 plus two?
We will begin by examining the
original sine function graph, before any transformations are applied. We notice the five key points
spanning one period of the sine curve. It appears that the scaling on the
𝑥-axis is in radians. And we know that sine has a period
length of two 𝜋 radians, which is approximately 6.28 radians. The main sine curve also has an
amplitude of one, which is the distance from the center line to the peak or the
trough.
We will now review the four
transformations of the sine function graph, beginning with vertical and horizontal
translations. A vertical translation, or shift,
of 𝑓 of 𝑥 by 𝑎 is 𝑓 of 𝑥 plus 𝑎. A horizontal translation, or phase
shift, of 𝑓 of 𝑥 by negative 𝑏 is 𝑓 of 𝑥 plus 𝑏. To clarify what is meant by this
rule, let’s look at two examples. 𝑓 of 𝑥 plus eight is a
translation left eight, and 𝑓 of 𝑥 minus eight is a translation right eight.
The sine graph is stretched
parallel to the 𝑦-axis by a scale factor of 𝑐 if 𝑓 of 𝑥 is multiplied by 𝑐. We may also refer to this as an
amplitude change. The sine graph is stretched
parallel to the 𝑥-axis by a scale factor of one over 𝑑 if we have 𝑓 of 𝑑 times
𝑥. We may also refer to this as a
period change. We will now clear some space in
order to determine which transformation is being applied to the sin of 𝑥 function
to get sin of 𝑥 plus two and how each of the five graphs given compares to the sin
of 𝑥 function.
If sin of 𝑥 equals 𝑓 of 𝑥, then
sin of 𝑥 plus two equals 𝑓 of 𝑥 plus two, where the 𝑏-value is two. Therefore, we are looking for the
sine graph with a horizontal translation of negative two, which does not affect the
original period of two 𝜋 or the amplitude of one. This transformation is just a
simple shift of two radians to the left.
Now we are ready to consider the
graph from option (A). If we highlight one period of the
given sine curve, we notice the amplitude is one, but the period has changed from
approximately 6.28 to approximately 3.14, or in other words from a period of two 𝜋
to a period of 𝜋. This curve appears to be the graph
of 𝑓 of two 𝑥, which effectively cuts the period in half. We also do not observe any
horizontal shift. Therefore, we eliminate option
(A).
Now we are ready to consider option
(B). If we again highlight one period of
the sine curve, we notice the amplitude is one. One period of the curve extends
from negative two to just past positive four. So this graph appears to have the
desired period length of two 𝜋, or approximately 6.28. And compared to the main sine
graph, which passes through the origin zero, zero, this curve passes through the
point negative two, zero, which is evidence of the horizontal shift of negative
two. So we conclude that option (B) is
the graph of 𝑓 of 𝑥 plus two.
Let’s take a quick look at the
other options to make sure that we have not missed anything. Here is the sine curve given as
option (C). This graph appears to have the
right period, but the amplitude is two instead of one, and there is no horizontal
translation of two either. Because of the amplitude change,
this appears to be the graph of two times 𝑓 of 𝑥. So we can confidently eliminate
option (C).
The graph given in option (D) is
noticeably different from any of the sine graphs we have seen so far because this
sine curve does not touch the 𝑥-axis at all. This curve still has a period of
two 𝜋 and an amplitude of one, but the most prominent difference is the vertical
translation up two. So this is a graph of 𝑓 of 𝑥 plus
two outside of the parentheses, not to be confused with 𝑓 of 𝑥 plus two inside the
parentheses. Since this graph is translated
vertically instead of horizontally, we can eliminate option (D).
The last graph to check is given by
option (E). This sine curve has the correct
amplitude and period, but it appears to be shifted to the right two, rather than to
the left two. So this is the graph of 𝑓 of 𝑥
minus two. It is important to remember with
horizontal translations, 𝑥 plus two moves the graph to the left and 𝑥 minus two
moves the graph to the right. So we eliminate option (E).
In summary, only option (B) showed
a sine graph with a period of two 𝜋, an amplitude of one, and a horizontal
translation of negative two. Therefore, option (B) is the graph
of 𝑦 equals sin of 𝑥 plus two.
