Function Families Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

In the family

y=b^x

(exponential), how does the graph change as

b

varies? Compare

b=2

b=1

b=0.5

x=2

1
$b=2$ : $2^2=4$ (growth). $b=1$ : $1^2=1$ (constant function $y=1$ ). $b=0.5$ : $(0.5)^2=0.25$ (decay).
2
For $b>1$ : exponential growth; $b=1$ : constant; $0<b<1$ : exponential decay. The base $b$ entirely determines growth/decay behavior.

Answer

b=2

y=4

(growth);

b=1

y=1

(constant);

b=0.5

y=0.25

(decay)

The exponential family

y=b^x

illustrates how a single parameter (the base

b

) creates fundamentally different behaviors. This family includes both growth and decay, with

b=1

as the degenerate constant case.

About Function Families

A function family is a group of functions sharing the same general form and behavior, differing only in the values of one or more parameters.

The family of quadratics [formula] (with [formula]) all share the same vertex at the origin. Describ

The family [formula] has a vertical asymptote at [formula] for each parameter [formula]. Analyze the

The family [formula] for integer [formula] has different behavior depending on whether [formula] is