We can see that . This tells us that .
There is an open dot on the graph of when . There will be a corresponding open dot on the graph of when .
sliding the graph
We begin with the function . The domain values of are represented by . We then define a new function, . The domain values of are represented by .
and are connected. To evaluate at , you evaluate at , where is some constant. Therefore, are -values. We have . This tells us what to do to to get . However, that is not how our story is told. Our story begins with and then is defined from .
We want to know how to get from .
To get corresponding values of from values of , subtract .
The graph of is obtained from the graph of by subtracting from domain values of , which appears to be reverse of the definition, . That is because the definition tells how to get the old domain for , rather than getting the new domain for .
Graph of .
Define a new function by , with the implied domain.
Which graph below is the graph of ?
To figure this out, we need to know how to get the new variable from the old
variable .
These tell us that and the graph of is the graph of shifted left .
The shape of the graph didn’t change. It just slid to the left. There are still three
pieces.
Shifting doesn’t change the shape of the graph or any of the relative measurements. It is a rigid movement.
Graph of .
Let’s create a new function called , which is a shifted sine function.
or
Labelling the horizontal axes and makes it easier to compare.
Graph of .
is periodic with a period of . If it is shifted by or any integer multiple of , then the resulting function is again .
If is shifted left by , then we get .
Graph of .
This agrees with the unit circle. As you move along the unit circle, the right/vertical coordinate (sine) has the same value as the left/horizontal coordinate (cosine) back a quarter-circle.
Graph of .
What does the graph of look like?
This is a shift from the basic absolute value graph. All we really need to know is where is the corner.
The corner occurs when the inside of the absolute value equals .
when . That is where the new corner sits.
Much of graphing follows this example.
There are important/strategic points for the function’s graph. You identify the position of those points. Then, the shifted graph follows the basic shape of the original graph.
For example, [ including ] has a vertical asymptote when the inside of the logarithm equals . The zero, and corresponding horizontal intercept, occurs when the inside equals .
Here is the graph of .
The graph looks the same as the basic logarithm graph, just slid left .
Let be a function with domain .
Below is the graph of .
The function is defined as with the implied domain.
We can see that . This tells us that .
There is an open dot on the graph of when . There will be a corresponding open dot on the graph of when .
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more examples can be found by following this link
More Examples of Shifting