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hi good people!.. smiley

 

I am given \(g(x)=a.{b^x}+q\)

 

points on the graph are \(C(0;1)\) and \(E(-1;{ -{1\over3}})\)

 

Determine a, b and q.

 

Okay, I know q is the y-intercept, so that is 1

so, now we plug in the values of any point to calculate b.

If the "b" was alone, I would be able to do it, but there is an "a" multiplied with the "b" as well....uhm, I'm slightly confused...can anyone please teach me the way?..

 

I do thank you all!

 Feb 22, 2019
 #1
avatar+103709 
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\(1=a+q\\ q=1-a\)

 

 

\(\frac{-1}{3}=ab^{-1}+q\\ \frac{-1}{3}=\frac{a}{b}+q \qquad \qquad b\ne0 \\ \frac{-1}{3}=\frac{a}{b}+1-a\\ 3b(\frac{-1}{3})=3b(\frac{a}{b}+1-a)\\ -b=3a+3b-3ab\\ 0=3a+4b-3ab \qquad \text{a and b both equal 0 or}\\ -3a(1-b)=4b\\ 3a(b-1)=4b\\ a=\frac{4b}{3(b-1)} \qquad \qquad b\ne1\\ \)

 

If    b =1    then

\(g(x)=a+q\\ g(x)=a+1-a\\ g(x)=1 \)

but then point G is not on the curve so  b cannot equal 1

 

So we have

 

.\(q=1-a\\ a=\frac{4b}{3(b-1)}\\ where \;\; b\ne1 \;\;\;\; b\ne0\)

 

PLUS, I am not so sure that b can be negative........   

The graph would be very broken if b was negative, I am not sure how much that matters.

Maybe another mathematician would like to comment here. 

 

Anyway ,...There is no single solution for this.

 

Here is the graph.

 

https://www.desmos.com/calculator/bqqgr7eyms

 

Here is the graph.

 Feb 22, 2019
 #2
avatar+514 
+1

Hi Melody,

 

According to the handbook, "b" can never be negative. It can only be either a fraction between 0 and 1, or greater than 1. Okay, so this I accept. There is an example in the book on a similar sum, but it is not clear to me. By the way it looks, it appears the "a" is disregarded and "b" is calculated. I followed that aproach and found "b" to be \(1 \over2\),

I then substituted this also into the equation and found "a" to be -1. But I'm not sure if that was the right way to go about it?

juriemagic  Feb 22, 2019
edited by juriemagic  Feb 22, 2019
 #3
avatar+103709 
-1

But my graph shows that a can vary as well.   frown

 

https://www.desmos.com/calculator/bxstlnfsa5

Melody  Feb 22, 2019
 #4
avatar+514 
+1

yes, I saw that, thank you so much for your time...I do apreciate!..as always!.. smiley

juriemagic  Feb 22, 2019

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