3.2 Modeling with exponential functions

Introduction

Long-term behavior of exponential functions

The role of

In Desmos, define $g(t) = ab^t+c$ and accept the prompt for sliders for $a$, $b$, and $c$. Edit the sliders so that $a$ has values from $a = 5$ to $a = 50$, $b$ has values from $b = 0.7$ to $b = 1.3$, and $c$ has values from $c = -5$ to $c = 5$ (each with a step-size of 0.01). In addition, in Desmos let $P = (0, g(0))$ and check the box to show the label. Finally, zoom out so that the window shows an interval of $t$-values from $-30 \le t \le 30$.

Set $b = 1.1$ and explore the effects of changing the values of $a$ and $c$. Write several sentences to summarize your observations.

$p(t) = 4372 (1.000235)^t + 92856$

$q(t) = 27931 (0.97231)^t + 549786$

$r(t) = -17398 (0.85234)^t$

$s(t) = -17398 (0.85234)^t + 19411$

$u(t) = -7522 (1.03817)^t$

$v(t) = -7522 (1.03817)^t + 6731$