Question

The table displays the corresponding y values for a given set of X values x value negative 1 y value 3 x value 0 y value 6 x value 1 y value 12x value 2 y value 24 x value 3 while value 48 create an exponential model for the data set shown in the XY chart

Answer 1

To create an exponential model from the given data points, we need to formulate the equations based on the typical form of an exponential function, which is:

\[ y = a \cdot b^x \]

Where:
- \( y \) is the dependent variable,
- \( x \) is the independent variable,
- \( a \) is the initial value (the value of \( y \) when \( x = 0 \)),
- \( b \) is the growth factor.

Based on the provided data:
- \( (x, y) = (-1, 3) \)
- \( (0, 6) \)
- \( (1, 12) \)
- \( (2, 24) \)
- \( (3, 48) \)

From the point where \( x = 0 \) and \( y = 6 \), we can deduce that \( a = 6 \), so the equation can be rewritten as:

\[ y = 6 \cdot b^x \]

Next, we can use the other points to solve for \( b \).

Using the point \( (1, 12) \):
\[ 12 = 6 \cdot b^1 \]
\[ b = \frac{12}{6} = 2 \]

Using the point \( (2, 24) \) to verify:
\[ 24 = 6 \cdot b^2 \]
Substituting \( b = 2 \):
\[ 24 = 6 \cdot 2^2 \]
\[ 24 = 6 \cdot 4 = 24 \] (This verifies.)

Using the point \( (3, 48) \) to validate:
\[ 48 = 6 \cdot b^3 \]
Substituting again:
\[ 48 = 6 \cdot 2^3 \]
\[ 48 = 6 \cdot 8 = 48 \] (This also verifies.)

Thus, the exponential model that fits the data set is:

\[ y = 6 \cdot 2^x \]

This is the equation that models the data points provided.