Question

evaluate independent practice
learning goal
i can construct exponential functions given a graph, a description of a relationship, or two input - output pairs.
lesson reflection (circle one)
starting getting there got it
complete the previous problems, check your solutions, then complete the lesson checkpoints below.
complete the lesson reflection above by circling your current understanding of the learning goal(s).
write an exponential function that includes the following points.

1. (-2,3) and (-1,9)
2. timothy’s salary, s(n), is a function of the number of years, n, that he has worked at his videogame company. the relationship between n and s(n) is shown in the table below.

number of years (n) salary s(n)
0 30,000
1 36,000
2 43,200
3 51,840
complete the table for the function using the given domain. then graph the function using the ordered pairs from the table.

1. f(x)=9(\frac{1}{3})^x with a domain of {0, 1, 2, 3, 4, 5}.

x f(x)=9(\frac{1}{3})^x
0
1
2
3
4
5

Explanation:

Step1: Recall exponential - function form

The general form of an exponential function is $y = ab^{x}$.

Step2: Substitute the given points into the function

For the point $(-2,3)$: $3=ab^{-2}=\frac{a}{b^{2}}$. For the point $(-1,9)$: $9 = ab^{-1}=\frac{a}{b}$.

Step3: Solve the system of equations

From $\frac{a}{b}=9$, we get $a = 9b$. Substitute $a = 9b$ into $\frac{a}{b^{2}}=3$. Then $\frac{9b}{b^{2}}=3$, which simplifies to $\frac{9}{b}=3$. Solving for $b$, we multiply both sides by $b$ and then divide by 3 to get $b = 3$. Substitute $b = 3$ into $a = 9b$, we find $a=27$. So the exponential function is $y = 27\cdot3^{x}$.

Answer:

$y = 27\cdot3^{x}$