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Mathematical Expression Editor
Vertical stretches and reflections
So far, we have seen the possible effects of adding a constant value to function output
and adding a constant value to function input . Each of these actions results in a
translation of the function’s graph (either vertically or horizontally), but otherwise
leaving the graph the same. Next, we investigate the effects of multiplication
the function’s output by a constant.
Given the parent function pictured
in below , what are the effects of the transformation for various values of
?
We first investigate the effects of and . For , the algebraic impact of this
transformation is that every output of is multiplied by . This means that the only
output that is unchanged is when , while any other point on the graph of the original
function will be stretched vertically away from the -axis by a factor of . We can see
this in image where each point on the original dark blue graph is transformed to a
corresponding point whose -coordinate is twice as large, as partially indicated by the
red arrows.
In contrast, the transformation is stretched vertically by a factor of , which has the
effect of compressing the graph of towards the -axis, as all function outputs of are
multiplied by . For instance, the point on the graph of is transformed to the graph
of on the graph of , and others are transformed as indicated by the purple
arrows.
To consider the situation where , we first consider the simplest case where in the
transformation . Here the impact of the transformation is to multiply every output of
the parent function by ; this takes any point of form and transforms it to , which
means we are reflecting each point on the original function’s graph across the -axis to
generate the resulting function’s graph. This is demonstrated in second graph where
is the reflection of across the -axis and will be discussed more in the next
section.
Finally, we also investigate the case where , which generates . Here we can think of
as : the effect of multiplying by first reflects the graph of across the -axis (resulting
in ), and then multiplying by stretches the graph of vertically to result in , as shown
in second graph .
We summarize and generalize our observations from the graphs above as
follows.
Given a function and a real number , the transformed function is a vertical stretch
of the graph of . Every point on the graph of gets stretched vertically to the
corresponding point on the graph of . If , the graph of is a compression of toward
the -axis; if , the graph of is a stretch of away from the -axis. Points where are
unchanged by the transformation.
Given a function and a real number , the transformed function is a reflection of
the graph of across the -axis followed by a vertical stretch by a factor of
.
Consider the functions and given in the following graphs.
a.
On the same axes as the plot of , sketch the following graphs: and . Be
sure to label several points on each of , , and with arrows to indicate
their correspondence. In addition, write one sentence to explain the overall
transformations that have resulted in and from .
b.
On the same axes as the plot of , sketch the following graphs: and . Be
sure to label several points on each of , , and with arrows to indicate
their correspondence. In addition, write one sentence to explain the overall
transformations that have resulted in and from .
c.
On the additional copies of the two figures below, sketch the graphs of the
following transformed functions: (at left) and . As above, be sure to label
several points on each graph and indicate their correspondence to points on the
original parent function.
d.
Describe in words how the function is the result of three elementary
transformations of . Does the order in which these transformations occur
matter? Why or why not?
What effect do the 1.5, 2, 0.5 and 0.25 seem to have?
(b)
Now disable the previous graphs and make sure that the following graphs are
enabled.
What effect do the 1.5, 2, 0.5 and 0.25 seem to have?
Let be a positive real number then the following transformations result in stretches
or shrinks of the graph Horizontal Stretches or Shrinks The transformation is a stretch by factor if
. The transformation is a shrink by factor if . Vertical Stretches or Shrinks The transformation is a stretch by factor if
The transformation is a shrink by factor if
Let . Find equations for the following transformations of .
(a)
is a vertical stretch of by a factor of .
(b)
is a horizontal shrink of by a factor of .
(a)
Transformation from to
(b)
Transformation from to
Reflections Across Axes
Points and are reflections of each other across the x-axis. Points and are
reflections of each other across the y-axis. In general, two points that are
symmetric with respect to a line are reflections of each other across that line.
The following transformations result in reflections of the graph of
Reflection across the x-axis
Reflection across the y-axis
Reflection through the origin
Notice that this is just a special case of horizontal or vertical stretching where the
factor we are multiplying by is !
Find an equation for the reflection of across each axis.
