Section 5.5 Reflections
Example 5.57. Warm-up.
Suppose
Now, let's define a new function
To fill in the top row, we need to find
We can fill out the rest of the table following the same logic:
Now, let's define a new function
Definition 5.64. Reflections/Flips.
There are two possible reflections (sometimes called flips):
Vertical Reflection: The graph of
is the same as the graph of but flipped vertically over the -axis.Horizontal Reflection: The graph of
is the same as the graph of but flipped horizontally over the -axis.
Example 5.65.
Below is the graph of
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We want to figure out which of the following is the graph of
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Since we are looking for
Checkpoint 5.66.
The graph of
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Sketch the graph of each of the following:
- To draw
we see that we are flipping the graph vertically (over the -axis). Since we are swapping the signs of all the -values, and times anything is still the points on the -axis (where ) stay where they are at. Next, we will move the corner points, then connect them together to get the whole graph. So, you would move the point on the original graph to Similarly, you would move the point on the original graph to move the point on the original graph to and move the point on the original graph to From there, you can connect the lines to get the final graph, shown in blue below, where the gray dotted line is the orignal graph of - To draw
we see that we are flipping the graph horizontally (over the -axis). Since we are swapping the signs of all the -values, and times anything is still the point on the -axis (where ) stays where it is at. Next, we will move the corner points, then connect them together to get the whole graph. So, you would move the point on the original graph to Similarly, you would move the point on the original graph to and move the point on the original graph to From there, you can connect the lines to get the final graph, shown in blue below, where the gray dotted line is the orignal graph of
Example 5.67.
Suppose
What if we wanted to flip it horizontally instead? In that case, we would be multiplying by
Notice how we simplified in the equation above. Any negatives inside of an even power cancel out and don't do anything. However, the negatives in the odd powers do not cancel, so they flip signs.
Checkpoint 5.68.
Suppose
Suppose
is the same as but flipped vertically. Write a formula forSuppose
is the same as but flipped horizontally. Write a formula for
-
Since we are flipping vertically, that means we are multiplying by
on the outside of the function. -
Since we are flipping horizontally, that means we are multiplying by
on the inside of the function.Notice how we simplified in the equation above. Any negatives inside of an even power cancel out and don't do anything. However, the negatives in the odd powers do not cancel, so they flip signs.
https://www.desmos.com/calculator/gnzbeb56rf