Waching daily Jul 23 2018

Given a function f(x), you can apply transformations to get a new function g(x).

The two types of transformations we will focus on are shifting and stretching.

In order to help make things concrete we're going to look at a specific function f(x)

known as the triangle wave.

The reason we chose this function is that it shares a lot of features with the sine

and cosine functions, but its reference points are integer values and it can be drawn using

straight lines.

Shifting transformations, also known as translations, come from adding or subtracting a number from

either the entire function or the argument of the function.

If it happens inside the argument, then it corresponds to a horizontal shift.

And if it happens to the entire function, it corresponds to a vertical shift.

Students often make mistakes because the horizontal transformations are opposite what we might

guess.

If c is a positive number then f(x-c) shifts the graph to the right and not to left.

To see why this happens, let's look at a chart of values for x and x-1.

Notice that the x-1 values are shifted to the right.

This means that when we evaluate the function those values will also be shifted to the right.

Vertical shifts don't have this behavior because they're transforming the graph after it has

read the values from the function.

So adding and subtracting values directly increases or decreases the final result.

Horizontal and vertical stretching comes from multiplying either the entire function or

the argument by some value.

Just as with shifting the horizontal stretching works opposite our intuitive guess.

Again, it comes from changing the values where you reading the values from when evaluating

the function.

Multiplying by 2 before evaluating the function means that you're reading values twice as

far from the origin.

If the value is negative, then we also reflect off the x or y axis depending on whether the

transformation is horizontal or vertical.

It is important that these transformations are always performed relative to the x-axis

to the y-axis and not relative to the curve itself.

This is the source of many mistakes when working on these problems.

The most general transformation we can get from these has the

form y = a f(bx - c) + d.

When graphing by transformations, the order in which the transformations are applied is

the following:

First, horizontal shift, then horizontal stretch, vertical stretch, and then vertical shift.

The order is related to the evaluation of the expression.

The reverse order for the horizontal parts matches with the pattern of things being backwards

relative to our intuition.

In the previous video, we graphed y = -2 sin(2x + pi/2) + 2.

We will now show how this looks when we graph it using transformations and see that both

end up with the same result.

We start with the graph of the sine function.

The graph is then shifted to the left by pi/2.

Then we stretch the graph horizontally by a factor of 1/2.

The negative 2 out front flips the graph vertically and then stretches it by a factor of 2.

And lastly, we shift it vertically by 2 units.

And we can see that this gives us the exact same result.

I encourage students to use key values instead of transformations for these functions because

it's easy to make mistakes on the horizontal transformations.

Students often anchor the horizontal stretch to a point on the graph instead of the y axis.

I have found that students find the key values to be a little bit more straightforward and

they make fewer mistakes.