Answer:
- red: f(2x+1) +3
- orange: (1/2)f(x-5)
- blue: -(f(x-6) -3
Step-by-step explanation:
You want the transformations of f(x) that correspond to the red, orange, and blue graphs given.
Transformation
Translation of a function's graph is accomplished by ...
f(x -h) +k . . . . . . f(x) translated right h units, up k units
Vertical expansion of the function's graph by a factor of k is accomplished by ...
k·f(x)
Note that k < 1 will compress the graph vertically.
Horizontal expansion of the function's graph by a factor of k is accomplished by ...
f(x/k)
When the value of k is less than 1, the graph is compressed. For example, compression by a factor of 2 (expansion by a factor of 1/2) will be accomplished by ...
f(x/(1/2)) = f(2x)
You will notice this is different from "your notes" in the problem statement.
Reflection over the x-axis requires all function values be negated:
-f(x)
Reflection over the y-axis requires all input values be negated:
f(-x)
Red graph
It is convenient to start by identifying a feature on the function's graph that can be found in the transformed graph. Here, there are two features that are useful: the curved vertex at the origin, and the pointed vertex at (-1, 1).
The amount of translation horizontally and vertically can be seen by where the curved vertex gets moved to. The curved vertex on the red graph is at (-1/2, 3), telling us it has moved -1/2 units horizontally and +3 units vertically.
At the same time we make this observation, we also note that the horizontal distance between the two identified vertices has been reduced to 1/2 unit, meaning the graph has been compressed horizontally by a factor of 2.
If we do the horizontal translation first, the amount of translation will be compressed horizontally along with other features of the graph. With that in mind, we have two ways to make the horizontal features correct:
- translate the graph left 1 unit, then compress it by a factor of 2.
- compress the graph by a factor of 2, then translate it left 1/2 unit.
The answer we have written above uses the first of these methods:
- replace x by (x +1) to translate left 1 unit: f(x+1)
- replace x by 2x to compress by a factor of 2: f(2x+1)
Of course, the translation upward by 3 units simply adds 3 to this transformation
red = f(2x+1) +3
Orange graph
The orange graph is compressed vertically by a factor of 2 (expanded by a factor of 1/2), and translated right 5 units. Using the transformations described above, we have ...
orange = (1/2)·f(x -5) . . . . . . reduced vertically, moved right 5
Blue graph
We notice the openings of this graph are upside down, meaning it has been reflected across the x-axis. This is accomplished by -f(x).
The result has been translated left 6 units and down 3, so we have to replace x with (x+6) and subtract 3 from the function value:
blue = -f(x +6) -3
__
Additional comment
For the red graph, if we did the horizontal compression first, we would have f(2x). Then the translation left 1/2 unit would be accomplished by replacing x with (x+1/2) to give f(2(x +1/2)). Simplifying inside parentheses would make that be f(2x +1), as above.
You need to be a little careful with reductions in size. Here, I have called the half-size graph a "compression by a factor of 2" or a "expansion by a factor of 1/2."
Other authors may use the wording "compression by a factor of 1/2" when the size multiplier is less than 1. They call any size reduction a "compression" by a fractional multiplier. There can be similar ambiguity in the use of the words "enlargement" and "reduction."
It can be helpful to use a graphing utility to make sure your function transformations give the desired result. A graphing calculator may be helpful for that.
