To determine the correct exponential equation that passes through the points (0, 6) and (1, 3), we can start with the general form of an exponential function:
\[ y = a \cdot b^x \]
where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)) and \( b \) is the base of the exponential function.
From the given point (0, 6):
At \( x = 0 \): \[ y = a \cdot b^0 = a = 6 \]
Now we know \( a = 6 \), so the equation becomes:
\[ y = 6 \cdot b^x \]
Next, we can use the second point (1, 3) to find \( b \):
At \( x = 1 \): \[ 3 = 6 \cdot b^1 \] \[ 3 = 6b \] \[ b = \frac{3}{6} = \frac{1}{2} \]
Now we can write the complete exponential equation:
\[ y = 6 \cdot \left(\frac{1}{2}\right)^x \]
Since \( \left(\frac{1}{2}\right)^x = 2^{-x} \), we can rewrite the equation as:
\[ y = 6 \cdot 2^{-x} \]
This function is not in the options provided. However, since none of the provided options directly match our derived equation, we can express \( 2^{-x} \) in terms of other bases to possibly find a match among the options.
From the options, we see:
- \( y = 6(12)^x \) → Increases very rapidly
- \( y = 6(2)^x \) → Increases rapidly, not matching points
- \( y = -2(6)^x \) → Decreases rapidly and negative
- \( y = -6(12)^x \) → Same as option 1 but negative
None of these options represent decreasing functions with a base less than 1, indicating \( (1/2)^x \). Therefore, none of the provided options are correct.
Thus, the answer aligns with the derived function but does not match any option from the list given.
1 answer