Question

interpret the function ( h(x)=3^{3x + 2} ) using the properties of exponents. drag one answer to each box. with a base of, the function in value for each 1 unit increase in the exponent. the expression ( 3^{3x+2} ) can be rewritten as. this means that the initial value is and the function by a factor of for each 1 unit increase in ( x ).

Explanation:

Step1: Use exponent rules

We know that \(a^{m + n}=a^{m}\times a^{n}\) and \((a^{m})^{n}=a^{mn}\). For \(h(x)=3^{3x + 2}\), we can rewrite it as \(3^{3x+2}=3^{2}\times3^{3x}\). And \(3^{3x}=(3^{3})^{x}=27^{x}\), so \(h(x)=9\times27^{x}\).

Step2: Analyze the base and growth

For an exponential function \(y = ab^{x}\), when \(b> 1\), the function is increasing. Here \(b = 27\). The general form of an exponential function is \(y=a\times b^{x}\), where \(a\) is the initial value (\(x = 0\) value) and \(b\) is the base. When \(x\) increases by \(1\), \(y\) changes by a factor of \(b\).

Answer:

With a base of \(27\), the function increases in value for each \(1\) unit increase in the exponent. The expression \(3^{3x + 2}\) can be rewritten as \(9\times27^{x}\). This means that the initial value is \(9\) and the function increases by a factor of \(27\) for each \(1\) unit increase in \(x\).