Problem: Using the knowledge of transformations guess the graph of the following function:
Solution: We start by identifying the basic function. We can
disregard the "+ 1" part, since this is just a shift that we can make, and we
can also disregard the operations performed with the argument,
multiplication by
So we start with the graph of cosine and first apply the transformations done
to the argument. There are two, in the order of calculation they are scaling
by
The scaling can be sometimes tricky, it might help to keep track of specific points. We pick some prominent point on the graph waiting to be shrinked, then we move it horizontally towards the y-axis so that its x-coordinate gets halved and we get its new position. If we do it with a few prominent points, we should get the shape right; this procedure may also serve as a check of the correctness of the outcome.
For instance, after we shift the graph and then flip it, the "hill"
that was originally in the middle moved to the position
Similarly, the first "dip" on the graph to the right from the origin was
(after shifting and flipping) at the position
Just to be sure, the second positive intercept of cosine with the
x-axis is when the argument of cosine is
If you are not sure, identify more such points and it should help you.
Anyway, now it is time to apply the shift to the value, which is simple, we just move the graph up by 1.
Note: Again, we can check on major features of this graph by comparing
it with results that we calculate. For instance, we know that the local
maxima of the graph should be at places where cosine is equal to 1. Cosine is
1 when its argument is of the form
Solving it we see that the hills should be at positions
Similarly we deduce that the "dips" should be at positions