Mathematics

3. Here is the graph of an exponential function [tex]f[/tex].

- Find an equation defining [tex]f[/tex].
- Explain your reasoning.

[tex]f(x) = 50 \cdot 2^x[/tex]

4. The equation [tex]f(t) = 24,500 \cdot (0.88)^t[/tex] represents the value of a car, in dollars, [tex]t[/tex] years after it was purchased.

a. What do the numbers 24,500 and 0.88 mean?

- 24,500: starting value
- 0.88: decay factor

b. What does [tex]f(9)[/tex] represent?

- [tex]f(9) = 24,500 \cdot (0.88)^9[/tex]

c. Sketch a graph that represents the function [tex]f[/tex] and shows [tex]f(0), f(1)[/tex], and [tex]f(2)[/tex].

5. The first two terms of an exponential sequence are 18 and 6. What are the next 3 terms of this sequence?

(From Unit 4, Lesson 1)

CC BY 2019 by Illustrative Mathematicians

Sure, let's solve the question about finding the next three terms of an exponential sequence where the first two terms are given as 18 and 6.

1. Identify the Known Values:
- First term ([tex]$a_1$[/tex]) = 18
- Second term ([tex]$a_2$[/tex]) = 6

2. Find the Common Ratio ([tex]$r$[/tex]):
- Since it is an exponential sequence, each term is found by multiplying the previous term by a constant, which is the common ratio.
- You can find [tex]$r$[/tex] using the formula for the second term:
[tex]\[
a_2 = a_1 \times r
\][/tex]
Substituting the known values:
[tex]\[
6 = 18 \times r
\][/tex]
- Solve for [tex]$r$[/tex]:
[tex]\[
r = \frac{6}{18} = \frac{1}{3}
\][/tex]

3. Calculate the Next Three Terms:
- Third term ([tex]$a_3$[/tex]):
[tex]\[
a_3 = a_2 \times r = 6 \times \frac{1}{3} = 2
\][/tex]
- Fourth term ([tex]$a_4$[/tex]):
[tex]\[
a_4 = a_3 \times r = 2 \times \frac{1}{3} = \frac{2}{3} \approx 0.67
\][/tex]
- Fifth term ([tex]$a_5$[/tex]):
[tex]\[
a_5 = a_4 \times r = \frac{2}{3} \times \frac{1}{3} = \frac{2}{9} \approx 0.22
\][/tex]

So the next three terms in the sequence are 2, approximately 0.67, and approximately 0.22.