Transformations - Pre-Calculus

The distance between the lines and is 6 units. Therefore, the first reflection should be 6 units below . Therefore, the first reflection is:

The distance between and is 7 units. Therefore, second reflection should be 7 units above .

The result is .

The result of reflecting across will become . Reflecting the equation across will become .

The correct answer is .

So far all of the transformations have come after or inside the function. But what about those that come before the function? This is the basis of reflections which can be made to reflect across the x or the y axis. Since a function takes an x value and returns a y value, the placement of our sign matters. For instance, is a parabola opening up whose vertex is at the origin. But if one was to place a - sign in front of the fucntion like this , then the function would always return a negative y value. Because of this, a negative sign in front of the function, but not "in" it will reflect over the x axis. This is equivalent to turning the first function,, a smile, into a frown. Use a graphing utility to visualize this. Now what if you placed a negative sign within the function? This would automatically make all x values negative. And since a function returns a y value and we will always be starting from the negative side of the x axis, this would reflect across the y axis. Use the function to visualize this in a graphing utility.

is the correct answer because a negative 1 inside the function (which is within the square root) shifts the function to the right (it is opposite of intuition) one unit and outside the function a negative 3 shifts the function 3 units down. Finally, because our negative sign is outside the function we will always yield negative y values and have thus reflected over the x axis.

First, if was reflected across , the new line will be .

Reflecting this line across will yield .

Reflecting across will yield .

The answer is .

Each graph you have ever seen is derived from a parent function which is the most basic form of that type of graph. For example, is a straight line across the real number c, is a straight diagonal line from bottom left to top right through the origin, and is a curved line starting from the origin and curving up and to the right (note this list is not all inclusive). Use a graphing utility to verify a few of these so you can see. When a function is not in its most basic parent form it has been shifted, reflected, stretched or compressed based on various changes to the parent function.

The basic rules for horizontal and vertical shifts are as follows:

if c is a positive real number, vertical and horizontal shifts in a function f(x) can be accompolished by:

1. vertical shift c units up

2. c units down

3. horizontal shift c units to the right

4. c units to the left

*note the opposite shift in the horizontal shifts does not follow the traditional number line.

will be shifted 3 units to the left (rule number 4) and then 3 units down (rule number 3) so its vertex will not be at the origin. will only be shifted up 3 units (rule number 1) from the origin. does not have a vertex since it is a straight line. is also a straight line, and it will not pass through the origin since it is shifted up 1 unit (rule number 1) . has a net movement of 0 units since the two shifts of 3 down and then 3 up cancel each other out (rule number 1) so its vertex will remain on at the origin and is therefore the correct answer.

Do not confuse the movement of an equation of a line with the movement of a point. Moving two units to the right has no effect on the equation of the horizontal line. Moving the equation two units down will change the horizontal line equation to .

Which of the following expressions corresponds to a vertical translation of f(x)

In my experience, vertical translations are the easiest way of changing a graph. A vertical translation will be of the form:

What this basically means is that we simply add our vertical translation onto the end of our function. This will simply make our function +c higher. In this case we need a vertical translation of positive 5. Do the following:

So our answer is:

Which of the following represents a horizontal shift of g(x) 3 to the right?

Horizontal translations take the form of:

The key thing to remember however, is that horizontal translations are a little counterintuitive. A positive "c" will move the function left, whereas a negative "c" will move the function to the right. (Maybe not what you'd expect)

So, if we are going to shift our function 3 to the right, we will do the following:

So we get:

Which of the following will shift any general function f(x) down 7 and left 16?

Recall that to shift any function vertically, we simply add or subtract a number on the end of the function. Because we are going down 7, we want to subtract 7 from the end. So far we have

Next, to translate horizontally, we need to add or subtract a number within f(x) itself. This means that wherever we have an x, we need to replace it with .

Further, we need to realize that horizontal translations work backwards from what you might expect. If we do , we will shift to the left. If we do , we will shift to the right.

So for this problem, we need to do , giving us:

The distance between the lines and is 6 units. Therefore, the first reflection should be 6 units below . Therefore, the first reflection is:

The distance between and is 7 units. Therefore, second reflection should be 7 units above .

The result is .

The result of reflecting across will become . Reflecting the equation across will become .

The correct answer is .

So far all of the transformations have come after or inside the function. But what about those that come before the function? This is the basis of reflections which can be made to reflect across the x or the y axis. Since a function takes an x value and returns a y value, the placement of our sign matters. For instance, is a parabola opening up whose vertex is at the origin. But if one was to place a - sign in front of the fucntion like this , then the function would always return a negative y value. Because of this, a negative sign in front of the function, but not "in" it will reflect over the x axis. This is equivalent to turning the first function,, a smile, into a frown. Use a graphing utility to visualize this. Now what if you placed a negative sign within the function? This would automatically make all x values negative. And since a function returns a y value and we will always be starting from the negative side of the x axis, this would reflect across the y axis. Use the function to visualize this in a graphing utility.

is the correct answer because a negative 1 inside the function (which is within the square root) shifts the function to the right (it is opposite of intuition) one unit and outside the function a negative 3 shifts the function 3 units down. Finally, because our negative sign is outside the function we will always yield negative y values and have thus reflected over the x axis.

First, if was reflected across , the new line will be .

Reflecting this line across will yield .

Reflecting across will yield .

The answer is .

Each graph you have ever seen is derived from a parent function which is the most basic form of that type of graph. For example, is a straight line across the real number c, is a straight diagonal line from bottom left to top right through the origin, and is a curved line starting from the origin and curving up and to the right (note this list is not all inclusive). Use a graphing utility to verify a few of these so you can see. When a function is not in its most basic parent form it has been shifted, reflected, stretched or compressed based on various changes to the parent function.

The basic rules for horizontal and vertical shifts are as follows:

if c is a positive real number, vertical and horizontal shifts in a function f(x) can be accompolished by:

1. vertical shift c units up

2. c units down

3. horizontal shift c units to the right

4. c units to the left

*note the opposite shift in the horizontal shifts does not follow the traditional number line.

will be shifted 3 units to the left (rule number 4) and then 3 units down (rule number 3) so its vertex will not be at the origin. will only be shifted up 3 units (rule number 1) from the origin. does not have a vertex since it is a straight line. is also a straight line, and it will not pass through the origin since it is shifted up 1 unit (rule number 1) . has a net movement of 0 units since the two shifts of 3 down and then 3 up cancel each other out (rule number 1) so its vertex will remain on at the origin and is therefore the correct answer.

Do not confuse the movement of an equation of a line with the movement of a point. Moving two units to the right has no effect on the equation of the horizontal line. Moving the equation two units down will change the horizontal line equation to .

Which of the following expressions corresponds to a vertical translation of f(x)

In my experience, vertical translations are the easiest way of changing a graph. A vertical translation will be of the form:

What this basically means is that we simply add our vertical translation onto the end of our function. This will simply make our function +c higher. In this case we need a vertical translation of positive 5. Do the following:

So our answer is:

Which of the following represents a horizontal shift of g(x) 3 to the right?

Horizontal translations take the form of:

The key thing to remember however, is that horizontal translations are a little counterintuitive. A positive "c" will move the function left, whereas a negative "c" will move the function to the right. (Maybe not what you'd expect)

So, if we are going to shift our function 3 to the right, we will do the following:

So we get:

Which of the following will shift any general function f(x) down 7 and left 16?

Recall that to shift any function vertically, we simply add or subtract a number on the end of the function. Because we are going down 7, we want to subtract 7 from the end. So far we have

Next, to translate horizontally, we need to add or subtract a number within f(x) itself. This means that wherever we have an x, we need to replace it with .

Further, we need to realize that horizontal translations work backwards from what you might expect. If we do , we will shift to the left. If we do , we will shift to the right.

So for this problem, we need to do , giving us:

The distance between the lines and is 6 units. Therefore, the first reflection should be 6 units below . Therefore, the first reflection is:

The distance between and is 7 units. Therefore, second reflection should be 7 units above .

The result is .

The result of reflecting across will become . Reflecting the equation across will become .

The correct answer is .