View Full Version : Exponential Functions

kasiviv002

06-18-2011, 10:53 PM

g(x)= ab^k(x-d) + c

In this what is the transformation of 'b'?

i know 'a' is a vertical stretch/compression

'k' is a horizontal stretch/compression

'd' is a shift to right/left

'c' is a shift up/down

So what is 'b'?

Ex. 3^x + 3

What is the transformation of b? i don't seem to understand it.

mmm4444bot

06-19-2011, 12:51 AM

Your function g above is a linear function. In other words, the graph of g(x) = a b^k (x - d) + c is always a straight line with slope a b^k.

Hence, the slope of the graph of g(x) increases as the parameter b gets bigger.

This cannot be what you're trying to discuss with all of those transformations, correct?

On a different topic, the exponential curve y = b^x changes at different rates, as the base b increases in value.

Increasing the value of the base causes the value of the power b^x to grow more slowly over the negative values of x and more quickly over the positive values of x.

Here is what happens to the shape of the graph of y = b^x, as the base increases from b = 2 to b = 10.

(Click image to expand)

[attachment=0:2q3gvcal]base.gif[/attachment:2q3gvcal]

BTW, your example of 3^x + 3 is no good because that expression does not fit the linear form a b^k (x - d) + c.

Are you mixing up different exercises into some sort of stew? :wink:

mmm4444bot

06-19-2011, 02:01 AM

:idea: I was walking the dogs, when suddenly I realized that you probably did not type grouping symbols around the exponent in your post. In other words, I'm thinking now that the exponent is k(x-d), instead of just k, yes?

Typing a b^k (x - d) + c means this:

a \cdot b^{k} \cdot (x - d) \;+\; c

Typing a b^[k(x - d)] + c means this:

a \cdot b^{k(x - d)} \;+\; c

If function g is supposed to be g(x) = a b^[k(x - d)] + c, then here is an example g(x) to show what happens to the graph as the base increases from b = 1/4 to b = 4 (with the following parameter values shown for a, k, d, and c).

g(x) = 4 b^(2(x - 1)) + 3 which simplifies to:

g(x) \;=\; 4b^{2x - 2} \;+\; 3

(Click image to expand)

[attachment=0:33m66533]exponential.gif[/attachment:33m66533]

Do you see that one of those bases causes the graph of g to be a horizontal line? What value for b between 1/4 and 4 do you think that is?

kasiviv002

06-19-2011, 02:20 PM

Yes sorry, i meant to type in a b^[k(x - d)] + c

The value for b that causes the graph to be a horizontal line would be 1 ?

So if b>1 then the exponential exponential increases from left to right

and if b<1 the exponential function decreases from left to right? Is this right?

mmm4444bot

06-19-2011, 03:43 PM

The value for b that causes the graph to be a horizontal line would be 1 ?

You got it. 1^x = 1 for all x, so a base of 1 makes g(x) a constant function.

if b>1 then the exponential [function] increases from left to right

and if b<1 the exponential function decreases from left to right

That is correct, but we should say that the function is decreasing for 0 ? b < 1 because b cannot be less than zero. 8-)

kasiviv002

06-19-2011, 05:12 PM

ok Thank you. I understand now :)

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