# Write Up 2

## In this write-up, I will explore vertical and horizontal shifts of the graph of

(This is problem 8 of Assignment 2.)

Let us consider equations of the form

What happens when we change d?

Example 1

Let's let e = 0 and change d.

As you can see, in the first equation, d = 0.

In the second equation, d = 2.

In the third equation, d = 4.

In the fourth equation, d = - 2.

In the fifth equation, d = - 4.

What do the graphs look like?

As you can see, each graph has the same shape as the graph of

but the graphs are shifted horizontally.

For example, when d = 2, the graph moves two units in the positive x-direction (to the right).

When d = 4, the graph moves four units to the right.

When d = - 2, the graph moves two units in the negative x-direction (to the left).

When d = - 4, the graph moves four units to the left.

Example 2

Let's let e = - 2 and change d.

As before, in the first equation, d = 0.

In the second equation, d = 2.

In the third equation, d = 4.

In the fourth equation, d = - 2.

In the fifth equation, d = - 4.

What do the graphs look like?

As before, each graph has the same shape as the graph of

but the graphs are shifted horizontally.

For example, when d = 2, the graph moves two units in the positive x-direction (to the right).

When d = 4, the graph moves four units to the right.

When d = - 2, the graph moves two units in the negative x-direction (to the left).

When d = - 4, the graph moves four units to the left.

Conclusion -- Changing d

In conclusion, changing d in the equation

results in a horizontal shift of the graph of

d units in the x-direction.  If d is positive, the shift is in the positive x-direction.  If d is negative, the shift is in the negative x-direction.

What happens when we change e?

Example 3

Let's let d = 0 and change e.

As you can see, in the first equation, e = 0.

In the second equation, e = 2.

In the third equation, e = 4.

In the fourth equation, e = - 2.

In the fifth equation, e = - 4.

What do the graphs look like?

As you can see, each graph has the same shape as the graph of

or

but the graphs are shifted vertically.

For example, when e = 2, the graph moves two units in the positive y-direction (up).

When e = 4, the graph moves up four units.

When e = - 2, the graph moves two units in the negative y-direction (down).

When e = - 4, the graph moves down four units.

Example 4

Let's let d = 2 and change e.  From above, we know that the graph of

will be shifted two units to the right of the graph of

Here are the equations that we will graph.

As before, in the first equation, e = 0.

In the second equation, e = 2.

In the third equation, e = 4.

In the fourth equation, e = - 2.

In the fifth equation, e = - 4.

What do the graphs look like?

As we predicted, each graph is shifted two units to the right of the graph of

All of the graphs have the same shape as the graph of

but the graphs are shifted vertically.

As before, when e = 2, the graph moves two units in the positive y-direction (up).

When e = 4, the graph moves up four units.

When e = - 2, the graph moves two units in the negative y-direction (down).

When e = - 4, the graph moves down four units.

Conclusion -- Changing e

In conclusion, changing e in the equation

results in a vertical shift of the graph of

e units in the y-direction.  If e is positive, the shift is in the positive y-direction.  If e is negative, the shift is in the negative y-direction.

Predicting what graphs will look like

Given an equation of the form

we can now predict what the graph will look like.

Suppose we are given the equation

In this equation, d = 3 and e = 4.  Therefore, we expect this graph to be the graph of

shifted three units in the positive x-direction and four units in the positive y-direction.  In other words, it will be shifted up four units and right three units.

Let's check!  Here is the graph:

We were right!  The graph is shifted to the right three units and up four units.

Suppose we are given the equation

In this equation, d = -1 and e = -3.  We expect this graph of this equation to be the graph of

shifted one unit in the negative x-direction and three units in the negative y-direction.  In other words, it will be shifted down three units and left one unit.

Let's check!  Here is the graph:

We were right again!  The graph is shifted to the left one unit and down three units.

Conclusion

In conclusion, if we are given an equation of the form

we know that the graph have the same shape as the graph of

but it will be shifted d units in the x-direction, and e units in the y-direction.

If d is positive, the graph will be shifted to the right, and if d is negative the graph will be shifted to the left.

If e is positive, the graph will be shifted up, and if d is negative, the graph will be shifted down.

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