Let's analyze the given table and determine whether a linear or exponential function best fits the data.

Given data:

x	y1	y2
1	3	7
2	9	14
3	27	28
4	81	56
5	243	112

It appears that there are two sequences of $y$-values corresponding to each $x$-value.

Step 1: Check if one of the $y$-values follows an exponential pattern:

• Looking at the first column of y-values:
  $3, 9, 27, 81, 243$ Each term is obtained by multiplying the previous term by 3: $3 \times 3 = 9, \quad 9 \times 3 = 27, \quad 27 \times 3 = 81, \quad 81 \times 3 = 243$ This follows the form: $y = a(b)^x$ Since the first value (when $x=1$) is $3$, we set $a = 3$ and $b = 3$, giving the function: $y = 3(3)^x$

Step 2: Check if the second column of y-values follows a linear pattern:

• The second sequence is: $7, 14, 28, 56, 112$ Checking the rate of change: $14 - 7 = 7, \quad 28 - 14 = 14, \quad 56 - 28 = 28, \quad 112 - 56 = 56$ Since the differences are doubling rather than remaining constant, this is not linear but exponential.

  The pattern follows: $7, 7(2), 7(2)^2, 7(2)^3, 7(2)^4$ This means: $y = 7(2)^{x-1}$

Final Answer:

1. First function (exponential):
   $y = 3(3)^x$
2. Second function (exponential):
   $y = 7(2)^{x-1}$

Would you like a step-by-step breakdown of how I found these functions?

Related Questions:

1. How do you determine if a function is linear or exponential?
2. Can you convert an exponential function to a logarithmic form?
3. What happens if an exponential function has a negative base?
4. How can you find an exponential function given only two points?
5. How do real-world applications use exponential functions (e.g., population growth, radioactive decay)?

Quick Tip:

To identify an exponential function, check if the ratio of consecutive terms is constant instead of their differences!