Question

1. open response - 2 points

mary is competing in a push up competition and wants to train for it. when she started training, she could do two push ups. by the first week, she could do six push ups and the week after that she could do eighteen push ups. assume this pattern of increase in push ups continues.
a. write the explicit formula to represent this scenario.
b. how many push ups will she do in week 6?

1. selected response - 1 point

an exponential function $f(x)$ has the following characteristics:

• $f(x)$ represents exponential growth
• there is a horizontal asymptote at $y = 1$
• the y-intercept is $(0, -2)$

which of the following is a possible equation for $f(x)$?
a. $f(x)=2\left(\frac{1}{2}\
ight)^x - 4$
b. $f(x)=3(2)^x - 5$
c. $f(x)= - 3(3)^x + 1$
d. $f(x)= - 2(4)^x + 1$

1. matching - 3 points

match each equation to its corresponding graph.
a. $f(x)= - 2(3)^x + 4$ b. $g(x)=2(3)^x + 4$ c. $h(x)=(3)^x + 4$
answers: graph ____ graph __ graph ____

Explanation:

Question 3

Step1: Identify sequence type

This is an arithmetic sequence where the first term $a_1=2$, common difference $d=18-16=2$.

Step2: Write explicit formula

Arithmetic sequence formula: $a_n = a_1 + (n-1)d$
Substitute values: $a_n = 2 + (n-1)2 = 2n$

Step3: Calculate week 6 value

Substitute $n=6$ into the formula: $a_6 = 2(6) = 12$

1. Exponential growth requires the base of the exponential term to be greater than 1.
2. Horizontal asymptote at $y=1$ means the constant term is 1 (for form $f(x)=ab^x + c$, asymptote is $y=c$).
3. Check y-intercept: substitute $x=0$ (since $b^0=1$) to verify $f(0)=-2$.

• For option d: $f(0)=-2(4)^0 +1 = -2+1=-2$, which matches. It has growth base $4>1$, asymptote $y=1$.

1. Graph 1: Shows exponential growth (increasing curve). $g(x)=2(3)^x +4$: base $3>1$, positive coefficient, so it grows; $y$-intercept $g(0)=2+4=6$, matches the increasing curve starting near $y=6$.
2. Graph 2: Shows exponential decay (decreasing curve). $f(x)=-2(3)^x +4$: base $3>1$, negative coefficient, so it decays; $y$-intercept $f(0)=-2+4=2$, matches the decreasing curve starting near $y=2$.
3. Graph 3: Shows exponential growth with smaller coefficient. $h(x)=(3)^x +4$: base $3>1$, coefficient 1 (smaller than 2 in $g(x)$), so slower growth; $y$-intercept $h(0)=1+4=5$, matches the moderate increasing curve starting near $y=5$.

Answer:

a. $a_n = 2n$
b. 12 push ups

---

Question 4