Question

select the best description of the following exponential equation.
$f(x) = 27(2.75)^x$
show your work here
hint: to add an exponent ($x^9$), type \exponent\ or press \^\
growing by 175% with every unit of $x$.
growing by 275% with every unit of $x$.
decaying by 275% with every unit of $x$.
decaying by 175% with every unit of $x$.

Explanation:

Step1: Recall exponential growth formula

The general form of an exponential growth function is \( f(x) = a(b)^x \), where \( a \) is the initial amount, and \( b \) is the growth factor. If \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay. The growth rate \( r \) is related to \( b \) by \( b=1 + r \) (for growth) or \( b = 1 - r \) (for decay), where \( r \) is a decimal.

Step2: Analyze the given function

For \( f(x)=27(2.75)^x \), the base \( b = 2.75 \). Since \( 2.75>1 \), it's a growth function.

Step3: Calculate the growth rate

Using \( b = 1 + r \), solve for \( r \): \( r=b - 1=2.75 - 1 = 1.75 \). To convert to a percentage, multiply by 100: \( 1.75\times100\% = 175\% \). So the function is growing by 175% with every unit of \( x \).

Answer:

Growing by 175% with every unit of \( x \) (the first option: "Growing by 175% with every unit of \( x \)")