Question

Autumn has a different plan. She thinks the $2 for practicing with a father is fine. However, she would make $6 if he makes his first basket, $18 when he makes 2 baskets, and so on, making her earnings triple with each subsequent basket made by her father.​

Answer 1

Answer:  6) 18, 54, 162, 486, 1458, 4374

                7) Exponential; because each previous term is multiplied by the growth factor (which is 3 "tripled" for this problem)

                8) see graph (attached)

Step-by-step explanation:

NOTE: Tripled is a hint that this is an exponential function which is of the form y = a(b)ˣ    where "a" is the initial value and "b" is the growth factor

For this problem, "a" is given as 2 and "b" is given as tripled which is 3

The equation is: y = 2(3)ˣ

Notice that they are using "m" to replace y and "b" to replace x

which results in the equation: m = 2(3)ᵇ

6)

$\begin{array}{c|l|l}\underline{\text{Baskets} (b)}&\underline{\qquad \qquad \text{Equation}\qquad\quad} &\underline{\text{Money Earned} (m)}\\0&m=2(3)^0&2\\1&m=2(3)^1\quad or \quad 2\times 3=&6\\2&m=2(3)^2\quad or \quad 6\times 3=&18\\3&m=2(3)^3\quad or \quad 18\times 3=&54\\4&m=2(3)^4\quad or \quad 54\times 3=&162\\5&m=2(3)^5\quad or \quad 162\times 3=&486\\6&m=2(3)^6\quad or \quad 486\times 3=&1458\\7&m=2(3)^7\quad or \quad 1458\times 3=&4374\\\end{array}$

7)

I know that the relationship is exponential because each previous term is multiplied by the growth factor which is 3 (tripled) for this problem.

8) see graph (attached)

Answer 2

We have to triple Autumn's earning for each basket, so the table would be

$\begin{array}{c|c}\text{Baskets}&\text{Earnings}\\0&2\\1&6\\2&18\\3&54\\4&162\\5&486\\6&1458\\7&4374\\\end{array}$

We expect an equation like

$m(b)=a\cdot x^b$

where $a$ is the initial value, and $x$ is the multiplier. Since we start with 2$, and we triple the amount with each basket, we have

$m(b)=2\cdot 3^b$

This relation is exponential, because every time a new basket is made, the previous amount is multiplied by a constant. In other words, we have

$\dfrac{m(b+1)}{m(b)}=3\quad \forall b\geq0$

For the graph, you just need to sketch the points from the table we draw above and connect them