Answer:
Shifting the graph of a function horizontally k units and shifting it vertically -k units do not have the same effect. Let me explain and include graphs to justify this.
When you shift a graph horizontally by k units, you replace x with (x - k) in the function. This means you're moving each point on the graph left by k units. On the other hand, when you shift a graph vertically by -k units, you replace f(x) with (f(x) - k), which means you move each point on the graph down by k units.
Here's a graphical representation to illustrate the difference:
Let's consider a simple linear function, f(x) = x, and perform both horizontal and vertical shifts:
Horizontal Shift by k units:
New function: f(x - k) = x - k
This shifts the graph of y = x to the right by k units.
Vertical Shift by -k units:
New function: f(x) - k = x - k
This shifts the graph of y = x down by k units.
Now, let's look at the graphs to see the difference:
Original graph (y = x):
markdown
Copy code
|
|
|
| *
| *
| *
|* (0,0)
---------------
|
Horizontal Shift by k units (y = x - k):
markdown
Copy code
| *
| *
| *
| *
| *
|* (k, 0)
---------------
|
Vertical Shift by -k units (y = x - k):
markdown
Copy code
|
|
|
| *
| *
| *
|* (0, -k)
---------------
|
As you can see, the horizontal shift by k units moves the graph to the right, while the vertical shift by -k units moves the graph down. These two shifts have different effects on the graph's position. So, shifting the graph horizontally k units is not the same as shifting it vertically -k units.
I am happy to help! :)