Which type of function has the form y = a*b^x with a base b > 0 and b ≠ 1?

Which type of function has the form y = a*b^x with a base b > 0 and b ≠ 1?

• A. Polynomial Function
• B. Exponential Function
• C. Linear Function
• D. Rational Function

Answer: (B)

When x shows up as an exponent, the function is exponential. In y = a·b^x, the output changes by a constant factor for each unit increase in x, because multiplying by b happens every time x increases by 1. That constant-multiple behavior is the defining feature of exponential growth or decay. The parameters a and b shape the graph: a vertically scales the graph, and b controls the rate of growth or decay, with b must be positive and not equal to 1 to avoid a constant function. If b>1 you get growth; if 0<b<1 you get decay. This form isn’t a polynomial, since polynomials have x raised to fixed integer powers and summed; it isn’t linear, which would be y = mx + c and has a constant additive rate of change; and it isn’t a rational function, which is a ratio of polynomials in x. So y = a·b^x with b>0 and b≠1 correctly describes an exponential function.

When x shows up as an exponent, the function is exponential. In y = a·b^x, the output changes by a constant factor for each unit increase in x, because multiplying by b happens every time x increases by 1. That constant-multiple behavior is the defining feature of exponential growth or decay. The parameters a and b shape the graph: a vertically scales the graph, and b controls the rate of growth or decay, with b must be positive and not equal to 1 to avoid a constant function. If b>1 you get growth; if 0<b<1 you get decay. This form isn’t a polynomial, since polynomials have x raised to fixed integer powers and summed; it isn’t linear, which would be y = mx + c and has a constant additive rate of change; and it isn’t a rational function, which is a ratio of polynomials in x. So y = a·b^x with b>0 and b≠1 correctly describes an exponential function.