Question

1. use the sequence for each of the following questions.

20, 9, 4.05, 1.8225...
a. write an exponential equation to model the sequence. hint: divide the 2nd term by the 1st to get your ‘b’ value.
b. graph the equation and identify the
y-intercept, domain, range, asymptote
y-int:
domain:
range:
asymptote:

Explanation:

Step1: Find the common ratio $b$

$b = \frac{9}{20} = 0.45$

Step2: Write the exponential equation

The general form of an exponential sequence is $a_n = a_1 \cdot b^{n-1}$, where $a_1=20$.
$a_n = 20 \cdot (0.45)^{n-1}$
For a continuous exponential function, we can rewrite it as $y = 20 \cdot (0.45)^x$ where $x$ is a non-negative integer for the sequence, and extended to real numbers for graphing.

Step3: Identify y-intercept

The y-intercept occurs at $x=0$.
$y = 20 \cdot (0.45)^0 = 20 \cdot 1 = 20$

Step4: Identify domain

For the exponential function (extended to real numbers), the domain is all real numbers. For the sequence, domain is positive integers, but the graph of the equation uses all real numbers.

Step5: Identify range

Since the base $0.45$ is positive and $20>0$, $y$ is always positive.

Step6: Identify asymptote

As $x \to +\infty$, $(0.45)^x \to 0$, so $y \to 0$.

Answer:

a. $a_n = 20(0.45)^{n-1}$ (for the sequence) or $y = 20(0.45)^x$ (for the continuous function)
b.
y-int: $(0, 20)$
Domain: All real numbers ($(-\infty, \infty)$)
Range: All positive real numbers $(0, \infty)$
Asymptote: $y=0$ (the x-axis)

(Note: To graph the equation, plot points like $(0,20)$, $(1,9)$, $(2,4.05)$, $(3,1.8225)$ and draw a smooth curve approaching $y=0$ as $x$ increases.)