The graph of \( y = a \cdot 2^x \) passes through the points $ (0, 3) $ and $ (2, 12) $. Find \( a \).
["Understanding the Exponential Graph: Finding the Constant ( a ) in ( y = a \cdot 2^x )", "When analyzing exponential functions, one essential step is identifying unknown constants using known data points. In this article, we’ll determine the value of ( a ) in the exponential function ( y = a \cdot 2^x ), given that the graph passes through the points ( (0, 3) ) and ( (2, 12) ).", "### The Exponential Function Form", "The general form of the exponential function under discussion is:\n[\ny = a \cdot 2^x\n]\nHere, ( a ) is the initial value of ( y ) when ( x = 0 ), and the base 2 determines the growth rate.", "### Step 1: Use the first point ( (0, 3) )", "Substitute ( x = 0 ) and ( y = 3 ) into the equation:\n[\n3 = a \cdot 2^0\n]\nSince ( 2^0 = 1 ), this simplifies to:\n[\n3 = a \cdot 1 \quad \Rightarrow \quad a = 3\n]", "So far, we find ( a = 3 ), but to verify, we must check this value using the second point.", "### Step 2: Validate with the second point ( (2, 12) )", "Substitute ( x = 2 ) and ( y = 12 ) into the equation using ( a = 3 ):\n[\ny = 3 \cdot 2^2 = 3 \cdot 4 = 12\n]\nThis matches the given ( y )-value exactly, confirming that ( a = 3 ) satisfies both points.", "### Conclusion", "By substituting the first point into the exponential model, we determined:\n[\na = 3\n]\nThis value ensures the function ( y = 3 \cdot 2^x ) passes through ( (0, 3) ) and ( (2, 12) ) correctly. Understanding how to solve for constants in exponential equations is crucial in fields ranging from finance to population modeling.", "---", "Keywords: exponential function ( y = a \cdot 2^x ), find ( a ), graph analysis, using points to determine constants, mathematical modeling.", "Meta Description: Solve for ( a ) in the exponential function ( y = a \cdot 2^x ) using the points ( (0, 3) ) and ( (2, 12) ). Learn how to find the growth factor ( a ) step by step."]
