# Effect of 'a' in y = e^(ax)

1. Apr 17, 2013

### 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.

2. Apr 17, 2013

### 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!

3. Apr 17, 2013

### k.udhay

Hi tiny-tim,

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

4. Apr 17, 2013

### tiny-tim

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

5. Apr 17, 2013

### 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???

6. Apr 17, 2013

### tiny-tim

uhh?

which way up are you?

7. Apr 17, 2013

### k.udhay

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

8. Apr 17, 2013

### 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?

9. Apr 17, 2013

### k.udhay

Y increases much faster than x.... :)

10. Apr 18, 2013

### rock.freak667

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

11. Apr 18, 2013

### Staff: Mentor

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.

12. Apr 18, 2013

### k.udhay

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 ex will have its tangent at Y = 1 at an angle of 45°. But why does one need this combination???

13. Apr 18, 2013

### k.udhay

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

14. Apr 19, 2013

### rock.freak667

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=ex → dy/dx = ex 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°

15. Apr 21, 2013

### k.udhay

Great explanation! Thank you!