Video Transcript
In this video, we will learn how to
translate or stretch the trigonometric function and find the rule of a trigonometric
function given the transformation. We will learn the notation for
horizontal and vertical translations and stretches. And we will learn the proper order
to apply a series of transformations on sine or cosine.
We’ll begin by recalling the key
features of the graphs of the main sine and cosine functions. When we plot the sine of an angle
against that angle, the result is the sine curve. Here, we have used degrees to mark
out the 𝑥-axis. But sometimes, we use radian
measures instead of degrees. We will need to be able to
recognize the graph of sine. So we must get familiar with the
key features, including the locations of the turning points and the 𝑥- and
𝑦-intercepts.
When we plot the cosine of an angle
against that angle, the result is the cosine curve. We notice that while the sine curve
has a 𝑦-intercept at zero, the cosine curve has a 𝑦-intercept at one. But, otherwise, the points follow
the same periodic shape. The main sine and cosine graphs
have some key features in common. They have the same domain of all
real numbers. And their range is the closed
interval from negative one to positive one. We also know that both sine and
cosine curves have an amplitude of one and a period of 360 degrees or two 𝜋
radians.
For our first example, let’s
practice identifying the graph of the cosine function with a single
transformation.
The figure shows the graph of 𝑓 of
𝑥. A transformation maps 𝑓 of 𝑥 to
𝑓 of 𝑥 minus two. Determine the coordinates of 𝐴
following this transformation.
We begin by recognizing the red
curve as the graph of the main cosine function. The features that clued us into
this fact are the amplitude of one, the period of 360, the 𝑦-intercept at one, and
the 𝑥-intercepts at 90 and 270. For this example, we need to
identify the effect of the transformation that maps 𝑓 of 𝑥 onto 𝑓 of 𝑥 minus
two. In particular, we will calculate
the effect on the coordinate point 180, negative one. The coordinates of point 𝐴
correspond to the fact that cos of 180 degrees equals negative one. So, we can find the new coordinate
by finding the value of 𝑓 of 𝑥 minus two at 𝑥 equals 180. That is cos of 180 minus two. And we know that cos of 180 equals
negative one. So, 𝑓 of 180 minus two equals
negative three. Therefore, the given transformation
maps 180, negative one to 180, negative three.
Let’s further explore the
implications of the transformation used in this example. Let’s create a table of values to
compare the output values of 𝑓 of 𝑥 compared to 𝑓 of 𝑥 minus two. For each point, we can see that the
value of 𝑓 of 𝑥 minus two is two less than the value of 𝑓 of 𝑥. If we plot this for all possible
values of 𝑥, we get the following graphs: in orange, cos of 𝑥 and in blue, cos of
𝑥 minus two. We have marked point 𝐴 on the
first graph and the new point 𝐵 on the graph after the transformation. This change causes point 𝐴 to
shift vertically downward by two onto point 𝐵. In fact, we can see the entire
graph has been shifted downward by two.
We can generalize this result in
the following way. For a real number constant 𝑑, 𝑓
of 𝑥 plus 𝑑 represents a vertical translation by zero, 𝑑 in the graph. In other words, the graph is
shifted upward by 𝑑. If 𝑑 is negative, as we just saw,
then the result is a downward shift in the graph. Now let’s consider what might
happen if we add or subtract a constant to the value of 𝑥 before substituting it
into the 𝑓 of 𝑥 function.
Let’s create a new table of values
to demonstrate the effect 𝑓 of 𝑥 minus 90 has on 𝑓 of 𝑥 equals cos of 𝑥. This particular transformation
tells us to subtract 90 from 𝑥 before evaluating cosine of that angle. So, we will add a third row to the
table, where 𝑥 minus 90 is calculated. Then, we evaluate cosine at the new
input values from row three. It may not be immediately obvious,
but the outputs in the bottom row compared to the outputs in the second row have all
been shifted to the right.
Let’s plot two curves to visualize
this transformation. As we can see, the graph has been
shifted to the right by 90. That is to say, by subtracting 90
from 𝑥 directly, the outputs move in the opposite direction. This result can also be
generalized. For a real number constant 𝑐, 𝑓
of 𝑥 plus 𝑐 represents a horizontal translation by negative 𝑐, zero in the
graph. In other words, the graph is
shifted left by 𝑐. If 𝑐 is negative, as we just saw,
then the result is a shift to the right.
Now, let’s summarize the main
function transformations. We will take a look at two types of
function transformations: translations and stretches. 𝑓 of 𝑥 plus a real number
constant 𝑑 represents a vertical translation by zero, 𝑑. 𝑓 of 𝑥 plus 𝑐 represents a
horizontal translation by negative 𝑐, zero. 𝑎 multiplied by 𝑓 of 𝑥
represents a vertical stretch by a scale factor of the absolute value of 𝑎. However, if 𝑎 is negative, the
function is first reflected in the 𝑥-axis. 𝑓 of 𝑏 multiplied by 𝑥
represents a horizontal stretch by a scale factor of the absolute value of one over
𝑏. And if 𝑏 is negative, then the
function is first reflected in the 𝑦-axis.
In our next example, we will
demonstrate how to find the coordinate of a point after a transformation using one
of these definitions.
The figure shows the graph of 𝑓 of
𝑥. A transformation maps 𝑓 of 𝑥 onto
𝑓 of two 𝑥. Determine the coordinates of 𝐴
following this transformation.
We begin by recognizing the red
curve as the graph of the main cosine function. The features that clued us into
this fact are the amplitude of one, the period of 360, the 𝑦-intercept at one, and
the 𝑥-intercepts at 90 and 270.
To answer this question, we must
identify the effect that the transformation mapping 𝑓 of 𝑥 onto 𝑓 of two 𝑥 has
on a single coordinate point. The point we are using is 𝐴, which
has coordinates 180, negative one. The coordinates of point 𝐴
correspond to the fact that cos of 180 degrees equals negative one. We recall that a function rule of
the form 𝑓 of 𝑏 multiplied by 𝑥 represents a horizontal stretch by a scale factor
of the absolute value of one over 𝑏. In this case, 𝑏 equals two. So, 𝑓 of two 𝑥 represents a
horizontal stretch by a scale factor of one-half. We know that a horizontal stretch
affects the 𝑥-coordinates only. In this case, each 𝑥-value is
multiplied by one-half.
Coordinate point 𝐴 has a value of
180 prior to the horizontal stretch. So, we take one-half of 180. Only vertical transformations
affect the 𝑦-coordinate. So the 𝑦-value in this case
remains unchanged. Therefore, according to the
horizontal stretch, the coordinates of the image of point 𝐴 are 90, negative
one. After applying the scale factor to
all the 𝑥-coordinates, we get a cosine curve with a period length that has also
been divided by two. We can check our answer by
substituting 𝑥 equals 90 into the 𝑓 of two 𝑥 function. Then, we get 𝑓 of 180. And we observe from the graph that
𝑓 of 180 equals negative one. This matches the 𝑦-coordinate of
the image of 𝐴 that we found. So we are confident in our final
answer.
So far, we have seen a single
transformation applied to each function. However, there will be occasions
where a function is mapped onto another function by a series of several
transformations. Generally, the order matters if the
transformations act in the same direction, in other words, when both transformations
have a horizontal effect or both have a vertical effect.
To better understand the importance
of sequencing transformations in the same direction, we will consider the functions
defined by two multiplied by the sin of 𝑥 plus one and two multiplied by
parentheses sin of 𝑥 plus one close parentheses. Since these functions are both
transformations of the main sine function, we start with a graph of one period of 𝑦
equals sin of 𝑥, drawn in pink. We obtain the first transformed
function by performing the vertical stretch by a scale factor of two, which causes
the 𝑦-coordinates to double, followed by a vertical translation up one, whereas the
second transformed function is obtained by first performing the vertical
translation, which increases the 𝑦-coordinates by one, followed by the vertical
stretch by a scale factor of two.
From this example, we notice that
when the order of the two vertical transformations changes, we have a slightly
different result. In order to avoid confusion, we
will follow a specific sequence when graphing transformations of sine or cosine. 𝑓 of 𝑥 maps onto 𝑎 times 𝑓 of
𝑏 multiplied by 𝑥 plus 𝑐 plus 𝑑 in the following order.
First, we would perform a vertical
stretch by a scale factor of the absolute value of 𝑎, where a negative 𝑎-value
results in a reflection in the 𝑥-axis. Then, we would perform a horizontal
stretch by a scale factor of the absolute value of one over 𝑏, where a negative
𝑏-value results in a reflection in the 𝑦-axis. Third, a horizontal translation is
given by negative 𝑐, zero. Fourth, a vertical translation is
given by zero, 𝑑.
Let’s apply this new information
about sequencing in our next example.
The figure shows the graph of 𝑓 of
𝑥. A transformation maps 𝑓 of 𝑥 to
four times 𝑓 of three 𝑥 minus 45 plus one. Determine the coordinates of 𝐴
following this transformation.
We begin by recognizing the red
curve as the graph of the main cosine function. The features that clued us into
this fact are the amplitude of one, the period of 360, the 𝑦-intercept at one, and
the 𝑥-intercepts at 90 and 270. Next, we recall the order in which
we sequence multiple function transformations from 𝑓 of 𝑥 to 𝑎 times 𝑓 of 𝑏
times 𝑥 plus 𝑐 plus 𝑑. First, we perform a vertical
stretch by a scale factor of the absolute value of 𝑎, where a negative 𝑎-value
results in a reflection in the 𝑥-axis. Second, we have a horizontal
stretch by a scale factor of the absolute value of one over 𝑏, where a negative
𝑏-value results in a reflection in the 𝑦-axis. Third, we have a horizontal
translation given by negative 𝑐, zero. And finally, we have a vertical
translation given by zero, 𝑑.
To identify the transformation that
maps 𝑓 of 𝑥 to four times 𝑓 of three 𝑥 minus 45 plus one, we rewrite four times
𝑓 of three 𝑥 minus 45 plus one as four times 𝑓 of three times parentheses 𝑥
minus 15 close parentheses plus one. And let 𝑎 equal four, 𝑏 equal
three, 𝑐 equal negative 15, and 𝑑 equal one. Then, 𝑓 of 𝑥 undergoes the
following transformations.
According to the 𝑎-value, we have
a vertical stretch by a scale factor of four. Since the 𝑎-value was positive,
there is no reflection in the 𝑥-axis. According to the 𝑏-value, we have
a horizontal stretch by a scale factor of one-third. Since the 𝑏-value is positive,
there is no reflection in the 𝑦-axis. According to the 𝑐-value, we have
a horizontal translation given by negative negative 15, zero, which equals 15, zero,
which means shifting right by 15. According to the 𝑑-value, we have
a vertical translation given by zero, one, which means the graph is shifted up by
one.
We can now apply the series of four
transformations to point 𝐴, which has coordinates 180, negative one. We know that vertical
transformations have an effect on the 𝑦-coordinate, whereas horizontal
transformations have an effect on the 𝑥-coordinate. Therefore, the vertical stretch by
a scale factor of four maps 180, negative one onto 180, negative one times four,
which equals 180, negative four. Then comes the horizontal stretch
by a scale factor of one-third, which maps 180, negative four onto 180 times
one-third, negative four, which equals 60, negative four. Then, we have a horizontal
translation that adds 15 to the 𝑥-coordinate, which gives us 75, negative four. And finally, the vertical
translation adds one to the 𝑦-coordinate, resulting in the coordinate point 75,
negative three.
Therefore, the coordinates of point
𝐴, following the given transformations, are 75, negative three. We can check our answer by
substituting 75 for 𝑥 into the transformed cosine function four times cos of three
𝑥 minus 45 plus one. After substituting 75 and
evaluating the expression, we find that the transformed function evaluated at 75
does indeed equal negative three, as expected.
We will finish by recapping some
key concepts from this video.
When applying a transformation to a
trigonometric function 𝑓 of 𝑥, 𝑎 times 𝑓 of 𝑥 represents a vertical stretch by
a scale factor of the absolute value of 𝑎, where a negative 𝑎-value results in a
reflection in the 𝑥-axis. 𝑓 of 𝑏 times 𝑥 represents a
horizontal stretch by a scale factor of the absolute value of one over 𝑏, where a
negative 𝑏-value results in a reflection in the 𝑦-axis. 𝑓 of 𝑥 plus 𝑐 represents a
horizontal translation given by negative 𝑐, zero, whereas 𝑓 of 𝑥 plus 𝑑
represents a vertical translation given by zero, 𝑑.
When a series of transformations
maps 𝑓 of 𝑥 onto 𝑎 times 𝑓 of 𝑏 times 𝑥 plus 𝑐 plus 𝑑, the four
transformations are applied in the given order: first, the vertical stretch, then
the horizontal stretch, then the horizontal translation, then the vertical
translation. We must also be familiar with the
key features of the graphs of the main sine and cosine functions before applying any
transformations. The most distinct differences
between the graphs of the main sine and cosine functions are their 𝑥- and
𝑦-intercepts, as shown in the orange and pink curves.
