Question

Which of these best represents the function shown in the table?

HINT: Using tables to identify Linear, Exponential or Quadratic Functions

(1 point)
Responses

f(x)=2(4x)

f of x is equal to 2 times open paren 4 to the x th power close paren

f(x)=x+2

f of x is equal to x plus 2

f(x)=6x+2

f of x is equal to 6 x plus 2

f(x)=x5

Answer 1

To determine which function best represents the data in the table, we need to assess whether the relationship described by the table can be identified as linear, exponential, or quadratic.

Here's a brief explanation of each type of function:

1. **Linear Functions**: These take the form \( f(x) = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. The output changes linearly as the input changes (the difference between outputs is constant for equal changes in inputs).

2. **Exponential Functions**: These take the form \( f(x) = a(b^x) \) where \( a \) is a constant, and \( b \) is the base of the exponential. The output changes multiplicatively as the input changes (the ratio between outputs is constant for equal changes in inputs).

3. **Quadratic Functions**: These take the form \( f(x) = ax^2 + bx + c \). The relationship shows a parabolic pattern and the second differences are constant.

Without the actual data from the table, we can still analyze the options based on their forms:

- **\( f(x) = 2(4^x) \)**: An exponential function.
- **\( f(x) = x + 2 \)**: A linear function.
- **\( f(x) = 6x + 2 \)**: Another linear function but with a different slope.
- **\( f(x) = x^5 \)**: A polynomial function, specifically a nonlinear function.

To select the best representation, review the changes in y values relative to x values in the table:

1. **If the changes in y values are consistent (same amount added for each step in x)**, a linear function is appropriate.

2. **If the ratio of y values for equal increments in x remains constant**, an exponential function is a better fit.

3. **If the second differences of y values are constant**, then a quadratic function is likely.

Since no table is provided, make your selection based on the patterns you observe:

- If you notice a consistent additive pattern, choose one of the linear functions (\( f(x) = x + 2 \) or \( f(x) = 6x + 2 \)).
- If you notice a pattern in multiplicative growth, select \( f(x) = 2(4^x) \).
- If the output increases more rapidly than linear (but not as fast as exponential), consider quadratic.

Based on what's provided, if we hypothetically assume a constant difference, \( f(x) = 6x + 2 \) could be representative of a linear relationship as the slope is steeper than \( f(x) = x + 2 \). Thus, if the data suggests linearity, I'd lean towards \( f(x) = 6x + 2 \) as representing a stronger linear relationship. However, check the given data to confirm your decision.