A curve and two labeled points on the curve are plotted in quadrant 1 of a coordinate plane.

Question

Given the graph of the exponential equation, write the correct exponential equation for the graph.
The points are labeled as: (0,3) and (1,4.5).

(1 point)
Responses

y=32(3)x
y equals Start Fraction 3 over 2 End Fraction left parenthesis 3 right parenthesis superscript x baseline

y=3(23)x
y equals 3 left parenthesis Start Fraction 2 over 3 End Fraction right parenthesis superscript x baseline

y=2(32)x
y equals 2 left parenthesis Start Fraction 3 over 2 End Fraction right parenthesis superscript x baseline

y=3(32)x
y equals 3 left parenthesis Start Fraction 3 over 2 End Fraction right parenthesis superscript x baseline
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Answer 1

To find the correct exponential equation given the points (0,3) and (1,4.5), we can use the general form of an exponential function:

\[ y = ab^x \]

where \( a \) is the initial value when \( x = 0 \), and \( b \) is the base of the exponential function.

1. From the point (0,3): \[ y = ab^0 = a \] Since \( y = 3 \) when \( x = 0 \), we have \( a = 3 \).

2. Now that we know \( a \), we can use the second point (1,4.5): \[ y = ab^x = 3b^1 = 3b \] Since \( y = 4.5 \) when \( x = 1 \), we have: \[ 3b = 4.5 \]

   Solving for \( b \): \[ b = \frac{4.5}{3} = 1.5 = \frac{3}{2} \]

Putting it all together, the equation is: \[ y = 3\left(\frac{3}{2}\right)^x \]

So, the correct choice from the given options is: y = 3(32)x This response reflects the proper form of the exponential function based on the two points provided.