Why is the dilation factor of a^kx

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In summary, the conversation discusses the concept of dilation and contraction of functions, specifically with regards to exponential functions. It is mentioned that a dilation factor of 1/k is present in the graph of a^kx, and the question is raised as to why it is not simply interpreted as k. An example is given to demonstrate the concept, with the conclusion that f(x/2) would be a dilation because it stretches out the x-axis by a factor of 2. The question is then reiterated, to which the response is that it was already explained in the previous post.
  • #1
autodidude
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...equal to 1/k? Why isn't it just interpreted as k?
 
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  • #2
Or is that just the way it is?
 
  • #3
I have absolutely no idea what you are asking. Can you please give an example?
 
  • #4
Had me baffled too, as OP talks about a specific function (e^x). But I think he's referring in general to calling f(x/a) a dilation of f(x). Not something I'm familiar with (by name) but I think I can guess what he's referring to.

Autodidude, say you have a function f(x) and you construct a new function g(x) = f(x/2). Can you see that if you plotted g(x) it would look identical to f(x) except that it would look like you'd stretched out the x-axis by a factor of two. Sketch some simple examples like f(x)=sin(x), g(x)=sin(x/2) and you'll soon see it. So f(x/2) would be a dilation because it has the same effect as stretching out the x-axis (by a factor of 2). On the other hand f(2x) would be a contraction. This is a general property of functions and not restricted to exponentials.
 
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  • #5
Graphs of exponential functions:

The graph of a^kx has a dilation factor 1/k parallel to the x-axis. So if it's f(x) = 2^2x, then the dilation factor is 1/2, I was curious as to why it isn't just 2
 
  • #6
autodidude said:
So if it's f(x) = 2^2x, then the dilation factor is 1/2, I was curious as to why it isn't just 2

There must be a problem with your browser not displaying things properly, because I'm pretty sure I just explained that in the previous post.
 

Related to Why is the dilation factor of a^kx

1. Why is the dilation factor of a^kx important in mathematical equations?

The dilation factor of a^kx is important because it represents the rate at which the graph of a function expands or contracts. This factor is crucial in understanding the behavior and transformations of a function.

2. How does the value of a affect the dilation factor of a^kx?

The value of a directly affects the dilation factor of a^kx. A larger value of a will result in a greater dilation factor, causing the graph of the function to expand more rapidly. Similarly, a smaller value of a will result in a smaller dilation factor and a slower expansion of the graph.

3. What does the exponent k in a^kx represent in terms of dilation factor?

The exponent k represents the degree of dilation in the x-direction. A positive value of k will cause the graph to expand, while a negative value of k will result in a contraction. The absolute value of k also affects the steepness of the graph.

4. How does changing the value of x affect the dilation factor of a^kx?

Changing the value of x will not directly affect the dilation factor of a^kx. However, it will affect the position of the graph on the x-axis. When x increases, the graph will shift to the left, and when x decreases, the graph will shift to the right. This can alter the overall appearance of the graph and the observed dilation factor.

5. Can the dilation factor of a^kx be negative?

Yes, the dilation factor of a^kx can be negative. This indicates a reflection of the graph across the y-axis. A negative dilation factor can also be achieved by using a negative value for a, which will result in a reflection and a change in the direction of the dilation.

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