SOLUTION: Below are two sets of data. One of the sets describes a linear function and the other describes an exponential function. Which is linear and which is exponential? Explain how yo

Question 138990: Below are two sets of data. One of the sets describes a linear function and the other describes an exponential function. Which is linear and which is exponential? Explain how you know. Find a formula for each function. For this problem, it is not acceptable to use results from your graphing calculator.
Table 1
x, g(x)
-2, 0.4444
-1, 1.3333
0, 4
1, 12
2, 36
3, 108
4, 324
5, 972
6, 2916
Table 2
x, f(x)
-2, 4
-1, 3.5
0, 3
1, 2.5
2, 2
3, 1.5
4, 1
5, 0.5
6, 0

Found 3 solutions by stanbon, Fombitz, vleith:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Below are two sets of data. One of the sets describes a linear function and the other describes an exponential function. Which is linear and which is exponential? Explain how you know. Find a formula for each function. For this problem, it is not acceptable to use results from your graphing calculator.
Table 1
x, g(x)
-2, 0.4444
-1, 1.3333
0, 4
1, 12
2, 36
3, 108
4, 324
5, 972
6, 2916
-------------
Looking at g(0) and g(1) you see there is a ratio of 12/4= 3
That ratio continues thruout the table.
So : Exponential with y = ab^x
Use (0.4) to find "a"; 4 = a^b^0; a = 4
So y = 4b^x
Use (1,12) to find "b"; 12 = 4b^1
b = 3
EQUATION: g(x) = 4*3^x
================================

Table 2
x, f(x)
-2, 4
-1, 3.5
0, 3
1, 2.5
2, 2
3, 1.5
4, 1
5, 0.5
6, 0
---------------
Look at f(1)/f(0)= 2.5/3 = (5/2/3) = 5/6
Look at f(2)/f(1) = 2/2.5 = 2/(5/2) = 4/5
No common ratio
--------------
Look at the successive difference: it is -1/2.
Eq. form y = mx + b
Since f(0) = 3, b= 3
Since y decreases (1/2) when x increases 1, m= -1/2
EQUATION: y = (-1/2)x + 3
================================

Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!
For a linear function the change in function, f(x), should be the same for each equal x division.
In other words, when you go from -2 to -1, the change in the function should be the same as going from 0 to 1 and 2 to 3 and so on.
g(-2)-g(-1)=0.4444-1.333=-0.8889
g(1)-g(2)=12-36=-24
Definitely not the same, definitely not linear.
Now look at f(x).
f(-2)-f(-1)=4-3.5=0.5
f(1)-f(2)=2.5-2=0.5
You can verify all of the other values to be sure.
f(x) is a linear function.
You can assume that g(x) is therefore the exponential function.
To be sure, you can do more analysis.
Exponential functions have the form,

where a and b are constants.

where c is a spacing constant, the distance between your x values.
Then you can look at the ratio of the values.

So, if your function is exponential, the ratio of two terms equals the same number.
g(-2)/g(-1)=0.4444/1.333=0.3333
g(1)/g(2)=12/36=0.3333
You can verify all of the other values to be sure.
g(x) is an exponential function.
Answer by vleith(2983)   (Show Source): You can put this solution on YOUR website!
Look at the way the value of f(x) changes with each change in x.
In table 1, each time x increases by 1, f(x) increases by a factor of 3. That is, each subsequent value for f(x) is 3 times greater than the previous value.
Since f(0) = 4, let's guess that
Now test that guess using other value of x. = 12. check. do some others to prove out the guess.

In table 2, each time x increases by 1, f(x) decreases by the quantity (0.5) . That is, each subsequent value for f(x) is 1/2 less than the previous value.
That is a line with a slope of -(1/2). The value of f(0)= 3. Which is the y intercept. So let's guess the line is
test that out using a few values for x and see how it goes.

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