Getting exponential function from table of x and y

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KmwdoaWNDW

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I've been given a table of x and y values that correspond to an exponential function.
The task is to write down the function and fill in the missing spaces in the table.
I tried asking my tutor for help, but he couldn't help me either.

FYI: I have not yet done anything with logarithms in class so there should be a solution without.

Here's the table:
x01212
y1210

Thanks in advance!
 
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What does an exponential function look like? Which columns in the table do you think are useful to figure out what the function is?
 
An exponential function is curved? I don't really know how to better explain the look of such a function in english, my apoligies.

Columns where x = 0 and x = 12 should be useful to get a rule for the function- at least I've seen someone on youtube make a rule for the function by for example, adding 1 to x in order to make the 2nd column match. This however does not work for this function as easily.

My tutor said something about using logartithms to solve this problem. According to him, it should be possible to get x where y=10 by using logsqrt(2)(10). However, I have no clue as to how to use logarithmic.


I've already solved a similar task where the table was exactly as in the question above, but the function is linear.

x01212108
y113/1214/12210
(posting this just because I've read that tutors should know the level of the student)
 
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It sounds like you may have made an attempt, but thought you might need logarithms to complete it. (You don't; but you will need a calculator.) Please show us the work you did, as far as you got; it may well be going in the right direction. Did you write an equation you have to solve? Let's see it.
 
There are two equivalent methods you can use.

An exponential function can be written \(\displaystyle y= Ce^{ax}\) for some numbers C and a. Since y= 1 when x= 0, \(\displaystyle 1= Ce^0= C\). Since y= 2 when x= 12, \(\displaystyle 2= Ce^{12a}\). Since C= 1, \(\displaystyle e^{12a}= 2\). Take the natural logarithm of both sides to solve for a.

You can also write the exponential function as \(\displaystyle y= Ca^x\) for some other numbers C and a. Since y= 1 when x= 0, \(\displaystyle 1= Ca^0= C\). Since y= 2 when x= 12, \(\displaystyle 2= Ca^{12}\). Since C= 1, \(\displaystyle a^{12}= 2\).
 
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Dr.Peterson said:
It sounds like you may have made an attempt, but thought you might need logarithms to complete it. (You don't; but you will need a calculator.) Please show us the work you did, as far as you got; it may well be going in the right direction. Did you write an equation you have to solve? Let's see it.
Click to expand...
I couldn't really do any real work or equations- I've only tried recreating an exponential function that almost crosses the points P1(0/1) and P2(12/2) : f(x)=1.0595^x. 1.0595 looks similar to something that my tutor got when he did something with logarithms.
 
Halls' second method is what I had in mind, which doesn't need a logarithm. What does it need instead?
 
KmwdoaWNDW said:
I couldn't really do any real work or equations- I've only tried recreating an exponential function that almost crosses the points P1(0/1) and P2(12/2) : f(x)=1.0595^x. 1.0595 looks similar to something that my tutor got when he did something with logarithms.
Click to expand...
That is the correct answer; so clearly you did SOME sort of work to get it. I asked for that work, to see what you are trying. How did you get that? Even if it was just trial and error, we can use that to help you learn. There is a fairly simple way to get it, so please work with us.
 
I went to geogebra.org and tried out some function that is exponential and crosses those 2 points.

But that value is just an approximation and with no equations or whatever I'd need to get it, I'd get like 0 points.
(trying to say that just writing "I used geogebra" is not enough as an answer.)
 
HallsofIvy said:
You can also write the exponential function as \(\displaystyle y= Ca^x\) for some other numbers C and a. Since y= 1 when x= 0, \(\displaystyle 1= Ca^0= C\). Since y= 2 when x= 12, \(\displaystyle 2= Ca^{12}\). Since C= 1, \(\displaystyle a^{12}= 2\).
Click to expand...

How do you know which ways you can write the exponential function? I only knew that \(\displaystyle x\) should be the exponent.
 
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KmwdoaWNDW said:
How do you know which ways you can write the exponential function? I only knew that \(\displaystyle x\) should be the exponent.
Click to expand...
You know how to write an exponential function by looking in your textbook for a definition! What have you been taught?

We've asked what your definition of exponential function is; and I asked to see some actual work. Just showing what kind of functions you tried would be an appropriate response to both requests. You clearly have some idea of a definition; what did you put into GeoGebra?

I'll suppose that your definition is [MATH]y = Ca^x[/MATH], where C and a are constants. But it could also have been just [MATH]y = a^x[/MATH], which is more restrictive, and appears to be what you used. Both, it turns out, will work.

Given that [MATH]f(0) = 1[/MATH], the first column of data, and putting the values into my form, you get [MATH]1 = Ca^0[/MATH]. This tells us that [MATH]C = 1[/MATH], so that if you started with the more restrictive form, it would still be correct.

Now we are also told that [MATH]f(12) = 2[/MATH], so that [MATH]2 = 1a^{12}[/MATH].

This is all in what HallsofIvy said, and this equation is what I expected you to have written. All that's left is to solve for a. Do you see how to do that?
 
Notice that we can write \(\displaystyle Ca^x= Ce^{ln(a^x)}- e^{x(ln(a))}\) so those two forms are equivalent.
 
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