Question

which function represents exponential growth? a) $y = 8x$ b) $y = 0.5x^2$ c) $y = 3(1.5)^x$ d) $y = -3(0.5)^x$

Explanation:

Step1: Recall exponential growth formula

The general form of an exponential function is \( y = ab^x \), where \( a>0 \) and \( b > 1 \) for exponential growth (since when \( b>1 \), as \( x \) increases, \( y \) increases exponentially). If \( 0 < b < 1 \), it's exponential decay.

Step2: Analyze Option A

The function \( y = 8x \) is a linear function (degree 1), not exponential. So A is not exponential growth.

Step3: Analyze Option B

The function \( y=0.5x^2 \) is a quadratic function (degree 2), not an exponential function. So B is not exponential growth.

Step4: Analyze Option C

For \( y = 3(1.5)^x \), we have \( a = 3>0 \) and \( b=1.5>1 \). This fits the exponential growth form \( y = ab^x \) with \( b > 1 \).

Step5: Analyze Option D

For \( y=-3(0.5)^x \), \( a=-3<0 \) (negative coefficient) and \( b = 0.5<1 \) (exponential decay factor). So it's not exponential growth.

Answer:

C) \( y = 3(1.5)^x \)