Which of the following represents a vertical shift up 5 units of f(x)?

A vertical translation can be accomplished by adding the desired amount onto the end of the equation. This means that f(x)+5 will shift f(x) up 5 units.

Which of the following represents a horizontal transformation of v(t) 3 units to the right?

To perform a horizontal transformation on a function, we need to add or subtract a value within the function, which looks something like this:

Now, counter intuitively, when we shift right, we will subtract. If we wanted to shift left, we would add.

So, to shift 3 to the right, we need:

There are four fundamental transformations that allows us to think of a function  as a transformation of a function , 

In our case,  and , so the width and/or height of our function will not change in the coordinate plane. 

We have  and . The number  will shift the function up  units along the -axis on the coordinate plane. The number  will shift  unit to the right on the coordinate plane. 

The correct answer is . Inside the portion being squared the distance moved is opposite the sign and is horizontal. Outside the squared portion the distance moved follows the sign (plus is up and minus is down) and is vertical.

For example the incorrect answer  would have its vertex at (1,-5).

To shift a polynomial  to the right by 2, we must replace x with x-2 in whatever the expression for the polynomial is. The logic of this is that every x value has a y value associated with it, and we want to give every x value the y value associated with the point that is 2 before it.

So, to get our shifted polynomial, we plug in x-2 as noted.

and then we combine like terms:

As a general rule, if you have a function , then in order to reflect across the x-axis, we compute , and in order to reflect across the y-axis, we compute . In our case, we are asked to compute the latter.

So, if , then .

To compute a reflection about the x-axis, calculate , and to calculate a reflection about the y-axis, calculate . To compute a reflection about the origin, simply combine both reflections into .

In our case, .

So, 

Let . In terms of ,

, being a logarithmic function, has a graph without a horizontal asymptote. As  represents the result of transformations of , it follows that its graph does not have a horizontal asymptote, either.

Let . In terms of ,

The graph of  has as its vertical asymptote the line of the equation . The graph of  is the result of three transformations on the graph of - a right shift of 3 units (  ), a vertical stretch (  ), and a downward shift of 2 units (  ). Of the three transformations, only the right shift affects the position of the vertical asymptote; the asymptote of  also shifts right 3 units, to .

, so the graph of  is the result of performing the following transformations:

1)  is the result of translating this graph two units right.

2)  is the result of reflecting the new graph about the -axis.