Exploring the Effects of Constant 'a' in the Exponential Function y = e^(ax)

[k.udhay]

Hi,
In the equation " y = e^(ax), what is the effect of constant 'a'. Like, what happens to the shape of the curve when it becomes -, +, high or low? Thanks.

 

[tiny-tim]

hi k.udhay!

(try using the X² button just above the Reply box )

tell us what you think, and then we'll comment!

 

[k.udhay]

Hi tiny-tim,

I don't see any X^2 button... Thanks.

 

[tiny-tim]

I don't see any X^2 button... Thanks.

you'll see it if you click the "Quote" button or the "Go Advanced" button

 

[k.udhay]

Hi tiny-tim,

About X², understood the point. :)
Well, what I can derive easily is when x turns 0, the curve is a straight vertical line. For '+' x, it lies on right hand side, for '-' side the curve travels in left hand side.

Ah... Now, higher the 'x' value, more it will become flat towards right... Am I right?

 

[tiny-tim]

Well, what I can derive easily is when x turns 0, the curve is a straight vertical line.

uhh?

which way up are you? ​

 

[k.udhay]

I am really sorry... It should be a horizontal line having an 'Y' interception 1... Correct now?

 

[rock.freak667]

Well let's take x=1.

for a=1, y=e^1
a=2, y=e^2
a=3, y=e^3

So what is happening to the value of y as 'a' increases?

 

[k.udhay]

Y increases much faster than x... :)

 

[rock.freak667]

Y increases much faster than x... :)

So can you now see the effect on the graph if 'a' is increased or decreased?

 

[Mark44]

Assuming you have the graph of y = f(x), here are a few variants.

• The graph of y = f(x) + a is a vertical translation (or shift) of the graph of y = f(x). If a > 0, the shift is upward. If a < 0, the shift is downward.
• The graph of y = f(x - a) is a horizontal translation (or shift) of the graph of y = f(x). If a > 0, the shift is to the right. If a < 0, the shift is to the left. For example, the graph of y = (x - 2)² looks like the graph of y = x², but shifted horizontally to the right. Instead of the vertex being at (0, 0), the vertex in the shifted graph is at (2, 0).
• The graph of y = -f(x) is a reflection across the horizontal axis of the graph of y = f(x).
• The graph of y = f(-x) is a reflection across the vertical axis of the graph of y = f(x). For example, the graph of y = sin(-x) looks like the graph of y = sin(x), but reflected across the y axis.
• The graph of y = af(x) represents an expansion away from the horizontal axis if a > 1, and a compression toward the horizontal axis if 0 < a < 1. If a < 0, there is also a reflection across the x-axis.
• The graph of y = f(ax) represents a compression toward the vertical axis if a > 1, and a compression away from the vertical axis if 0 < a < 1. If a < 0, there is also a reflection across the y-axis.

 

[k.udhay]

So can you now see the effect on the graph if 'a' is increased or decreased?

Thanks rock.freak. Yeah, I think I can figure out the other cases. Now, let me ask my second question:
What is the need for finding this term 'e' which has a strange value of 2.718... When I read in wiki., it says a curve following e^x will have its tangent at Y = 1 at an angle of 45°. But why does one need this combination?

 

[k.udhay]

Assuming you have the graph of y = f(x), here are a few variants.

• The graph of y = f(x) + a is a vertical translation (or shift) of the graph of y = f(x). If a > 0, the shift is upward. If a < 0, the shift is downward.
• The graph of y = f(x - a) is a horizontal translation (or shift) of the graph of y = f(x). If a > 0, the shift is to the right. If a < 0, the shift is to the left. For example, the graph of y = (x - 2)² looks like the graph of y = x², but shifted horizontally to the right. Instead of the vertex being at (0, 0), the vertex in the shifted graph is at (2, 0).
• The graph of y = -f(x) is a reflection across the horizontal axis of the graph of y = f(x).
• The graph of y = f(-x) is a reflection across the vertical axis of the graph of y = f(x). For example, the graph of y = sin(-x) looks like the graph of y = sin(x), but reflected across the y axis.
• The graph of y = af(x) represents an expansion away from the horizontal axis if a > 1, and a compression toward the horizontal axis if 0 < a < 1. If a < 0, there is also a reflection across the x-axis.
• The graph of y = f(ax) represents a compression toward the vertical axis if a > 1, and a compression away from the vertical axis if 0 < a < 1. If a < 0, there is also a reflection across the y-axis.

Thank you, mark. Your exlanation will help me extimate the behaviour of curves very easily!

 

[rock.freak667]

Thanks rock.freak. Yeah, I think I can figure out the other cases. Now, let me ask my second question:
What is the need for finding this term 'e' which has a strange value of 2.718... When I read in wiki., it says a curve following e^x will have its tangent at Y = 1 at an angle of 45°. But why does one need this combination?

You can consider 'e' to just be another special constant without having to go into the details of how to define it. Much like π.

For the y=1, θ=45° thing, you don't really have to memorize something like that but you can derive that as follows:

y=e^x → dy/dx = e^x i.e. dy/dx =y

So that when y=1, dy/dx =1 i.e. the gradient of the tangent at y=1 is 1.

The angle a straight line makes with the x-axis of gradient 'm' is given by tanθ=m or θ=tan^-1(m) so in this case, m=1 such that θ=45°

 

[k.udhay]

You can consider 'e' to just be another special constant without having to go into the details of how to define it. Much like π.

For the y=1, θ=45° thing, you don't really have to memorize something like that but you can derive that as follows:

y=e^x → dy/dx = e^x i.e. dy/dx =y

So that when y=1, dy/dx =1 i.e. the gradient of the tangent at y=1 is 1.

The angle a straight line makes with the x-axis of gradient 'm' is given by tanθ=m or θ=tan^-1(m) so in this case, m=1 such that θ=45°

Great explanation! Thank you!