Exponential Essentials Assessment Document
Exponential Essentials
Algebra 1 • Unit Assessment
Student Name
Date
Period
Select the best answer for each of the following 20 questions. Ensure you show work for solving equations in the margins if needed.
1
Rewrite the function \(y = 10(0.65)^{t/8}\) to identify the percent rate of change. What is the approximate rate of decay?
A) 35%
B) 5%
C) 65%
D) 95%
2
Identify whether the function \(f(t) = 4(1.25)^{t+3}\) represents growth or decay and find the percent rate of change.
A) Decay; 25%
B) Growth; 1.25%
C) Growth; 25%
D) Decay; 75%
3
Which type of function does the following table represent?
<table class="w-64 border-collapse border border-slate-400 mb-4"><tbody><tr><th class="border border-slate-400 px-3 py-1 bg-slate-100">\(x\)</th><td class="border border-slate-400 px-3 py-1 text-center">0</td><td class="border border-slate-400 px-3 py-1 text-center">1</td><td class="border border-slate-400 px-3 py-1 text-center">2</td><td class="border border-slate-400 px-3 py-1 text-center">3</td></tr><tr><th class="border border-slate-400 px-3 py-1 bg-slate-100">\(y\)</th><td class="border border-slate-400 px-3 py-1 text-center">3</td><td class="border border-slate-400 px-3 py-1 text-center">6</td><td class="border border-slate-400 px-3 py-1 text-center">12</td><td class="border border-slate-400 px-3 py-1 text-center">24</td></tr></tbody></table>
A) Linear growth
B) Exponential growth
C) Exponential decay
D) Neither
4
A television's value is \$1500 and decreases by 14% each year. Which function represents the value \(y\) after \(t\) years?
A) \(y = 1500(1.14)^t\)
B) \(y = 1500(0.14)^t\)
C) \(y = 1500(0.86)^t\)
D) \(y = 14(1500)^t\)
5
You deposit \$750 in a savings account that earns 5% annual interest compounded quarterly. How many times per year is the interest calculated?
A) 1
B) 2
C) 4
D) 12
6
Solve the equation: \(5^x = 5^{3x - 2}\)
A) \(x = 1\)
B) \(x = -1\)
C) \(x = 2\)
D) \(x = 0\)
7
Solve the equation: \(\frac{1}{9} = 3^{x + 6}\)
A) \(x = -4\)
B) \(x = -8\)
C) \(x = -3\)
D) \(x = 2\)
8
Solve the equation: \(2^x = \frac{1}{128}\)
A) \(x = -7\)
B) \(x = -6\)
C) \(x = 7\)
D) \(x = 64\)
9
Solve the equation: \(256^{x + 2} = 16^{3x - 1}\)
A) \(x = 3\)
B) \(x = 5\)
C) \(x = -5\)
D) \(x = 2.5\)
10
Solve the equation: \(27^{2x + 2} = 81^{x + 4}\)
A) \(x = 2\)
B) \(x = 5\)
C) \(x = 4\)
D) \(x = 10\)
11
What is the common difference for the sequence shown in the table?
<table class="w-64 border-collapse border border-slate-400 mb-4"><tbody><tr><th class="border border-slate-400 px-3 py-1 bg-slate-100">\(n\)</th><td class="border border-slate-400 px-3 py-1 text-center">1</td><td class="border border-slate-400 px-3 py-1 text-center">2</td><td class="border border-slate-400 px-3 py-1 text-center">3</td><td class="border border-slate-400 px-3 py-1 text-center">4</td></tr><tr><th class="border border-slate-400 px-3 py-1 bg-slate-100">\(a_n\)</th><td class="border border-slate-400 px-3 py-1 text-center">-6</td><td class="border border-slate-400 px-3 py-1 text-center">8</td><td class="border border-slate-400 px-3 py-1 text-center">22</td><td class="border border-slate-400 px-3 py-1 text-center">36</td></tr></tbody></table>
A) \(d = 14\)
B) \(d = -14\)
C) \(d = 2\)
D) \(d = 6\)
12
Identify the explicit rule for the sequence: \(400, 100, 25, 6.25, ...\)
A) \(a_n = 400(4)^{n-1}\)
B) \(a_n = 400(0.25)^{n-1}\)
C) \(a_n = 100(4)^{n-1}\)
D) \(a_n = 400 - 300(n-1)\)
13
Find the recursive rule for the arithmetic sequence: \(a_1 = -6, d = 14\).
A) \(a_n = a_{n-1} + 14\)
B) \(a_n = 14a_{n-1}\)
C) \(a_n = a_{n-1} - 6\)
D) \(a_n = -6(14)^{n-1}\)
14
The first two terms of a geometric sequence are \(a_1 = 3\) and \(a_2 = -12\). What is the third term, \(a_3\)?
A) -27
B) 36
C) 48
D) -48
15
Atmospheric pressure \(P\) decreases with altitude \(a\) (in meters) and can be modeled by \(P = (0.99988)^a\). Identify the decay factor.
A) 1
B) 0.00012
C) 0.99988
D) -0.99988
16
Each day you run 20% farther than the previous day. On Day 1, you run 1 km. Which explicit rule represents the distance you run on Day \(n\)?
A) \(a_n = 1(0.20)^{n-1}\)
B) \(a_n = 1(1.20)^{n-1}\)
C) \(a_n = 20(1)^{n-1}\)
D) \(a_n = 1 + 0.20(n-1)\)
17
Solve the equation: \(3^{x-2} = 1\)
A) \(x = 0\)
B) \(x = 2\)
C) \(x = -2\)
D) \(x = 1\)
18
Solve the equation: \((\frac{1}{16})^{3x} = 64^{2(x+8)}\)
A) \(x = -2\)
B) \(x = -4\)
C) \(x = 4\)
D) \(x = -2.67\)
19
Rewrite \(y = (1.06)^{8t}\) as \(y \approx a(1+r)^t\). What is the annual growth rate (rounded to the nearest whole percent)?
A) 6%
B) 48%
C) 59%
D) 159%
20
What is the next term in the sequence: \(2, 6, 18, 54, ...\)?
A) 108
B) 72
C) 162
D) 216
