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Quizzes > General Knowledge & Mixed > Mixed Topic

Take the Unit 3, 4 & 5 Comprehensive Review Quiz

Ready for the ultimate unit 4 practice test and unit 5 review questions? Jump in now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style illustration showing numbered cards 3 4 5 on golden yellow background quiz badge icons and progress symbols

Ready to reinforce your learning? Dive into our free unit review quiz designed to test your mastery of Units 3, 4 & 5 concepts. Whether you're warming up with a unit 4 practice test or tackling unit 5 review questions, this comprehensive unit trivia journey will help you spot gaps, boost confidence, and prepare for any challenge. Ideal for students seeking a targeted unit 3 review quiz, you'll engage with varied question styles and real-world scenarios. Plus, kickstart your prep with our Unit 4 pre test quiz and refine spatial reasoning through a fun unit 5 geometry test . Ready to supercharge your skills? Let's begin now!

Which of the following relations represents a function?
{(1, 2), (2, 3), (3, 4)}
{(1, 2), (1, 3), (2, 4)}
{(2, 2), (2, 3), (3, 5)}
{(3, 4), (4, 5), (3, 6)}
A relation is a function if each input has exactly one output. In the correct relation, each first element is unique. The other options repeat an input with different outputs. See more at Math is Fun: Functions.
What is the domain of f(x) = 1 / (x - 3)?
(-?, 3) ? (3, ?)
(-?, 1) ? (1, ?)
All real numbers
(3, ?)
A rational function is undefined where its denominator is zero. Here, x - 3 = 0 at x = 3, so x cannot equal 3. Thus the domain excludes 3. More details at Purplemath: Domain of a Rational Function.
What is the slope of the line passing through the points (2, 5) and (5, 11)?
2
3
1
-2
Slope is rise over run: (11 - 5) / (5 - 2) = 6 / 3 = 2. The other values do not match that calculation. See slope definition at Khan Academy: Slope of a Line.
Simplify log base 5 of 5³.
3
5
15
1/3
By definition, log?(5³) = 3 because 5³ is the power of 5. The other options confuse multiplication or inversion. More at Math is Fun: Logarithms.
Solve for x: log?(32) = x.
5
4
6
10
Since 2? = 32, log?(32) = 5. Other choices reflect incorrect exponent values. Review at Khan Academy: Logarithm Review.
What is the domain of y = ln(x - 1)?
(1, ?)
(-?, 1)
(0, ?)
All real numbers
The natural log requires its argument to be positive: x - 1 > 0 ? x > 1. Hence the domain is (1, ?). See restrictions at Purplemath: Domain of Logarithms.
What is sin(?/6)?
1/2
?3/2
1
?2/2
On the unit circle, ?/6 corresponds to 30°, where the sine value is 1/2. The other values correspond to different special angles. See reference at Math is Fun: Unit Circle.
What is the period of y = cos(x)?
2?
?
360°
?/2
The cosine function completes one full cycle over an interval of 2? radians. ? would be half a cycle. More at Khan Academy: Trig Function Graphs.
If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?
19
18
7
17
First compute g(3) = 9, then f(9) = 2·9 + 1 = 19. Composing in the other order gives a different result. See function composition at Purplemath: Function Composition.
The graph of y = f(x) is shifted 3 units up. What is the new function?
f(x) + 3
f(x) - 3
f(x + 3)
3·f(x)
Vertical shifts add or subtract constants outside the function: up by 3 becomes f(x) + 3. Inside shifts (x ± c) move left/right. More at Khan Academy: Graph Translations.
Solve for x: e? = 5.
ln(5)
log(5)
5·ln(e)
e?
Taking the natural log of both sides gives x = ln(5). log(5) without base is base 10. More at Math is Fun: Exponential Equations.
Solve the equation log(x) + log(x - 3) = 1 (base 10).
5
-2
10
-5
Combine logs: log[x(x - 3)] = 1 ? x² - 3x = 10 ? x² - 3x - 10 = 0. Solutions are 5 and -2, but x - 3 must be positive so x = 5 only. See process at Purplemath: Logarithm Equations.
Simplify tan²(x) + 1.
sec²(x)
csc²(x)
1/sec²(x)
-cos²(x)/sin²(x)
Using a Pythagorean identity: 1 + tan²(x) = sec²(x). The other forms correspond to different identities. More at Khan Academy: Trig Identities.
Solve for x in [0, 2?): cos(x) = 1/2.
x = ?/3 and 5?/3
x = ?/6 and 11?/6
x = 2?/3 and 4?/3
x = ?/4 and 7?/4
Cosine equals 1/2 at 60° and 300°, which are ?/3 and 5?/3 in radians. The other pairs correspond to sine or other values. Reference at Math is Fun: Unit Circle.
Convert 60° to radians.
?/3
2?/3
?/2
?/6
Multiply by ?/180: 60·?/180 = ?/3. The other options come from different degree measures. More at Purplemath: Degrees to Radians.
What is the effect on the graph of y = f(x) under the transformation y = f(-x)?
Reflection across the y-axis
Reflection across the x-axis
Translation left by 1 unit
Translation up by 1 unit
Replacing x with -x reflects the graph across the y-axis. An x-axis reflection uses -f(x). Shifts involve adding constants. More at Khan Academy: Function Reflections.
What is the inverse function of f(x) = 2x + 3?
(x - 3) / 2
(x + 3) / 2
2x - 3
1 / (2x + 3)
Swap x and y: x = 2y + 3 ? y = (x - 3)/2. Inverse functions reverse the original mapping. See method at Purplemath: Inverse Functions.
Solve for x: 2·3? = 18.
2
3
1
-2
Divide both sides by 2: 3? = 9, which gives x = 2 because 3² = 9. Other choices are incorrect exponents. More at Math is Fun: Exponential Equations.
What is the range of y = log(x + 4)?
All real numbers
y > 0
y < -4
y > -4
A logarithm outputs any real number as its exponent. There is no restriction on y. The domain of x + 4 > 0, but the range is all reals. See explanation at Purplemath: Domain & Range of Logs.
Which identity is equivalent to sin(2x)?
2 sin(x) cos(x)
sin²(x) - cos²(x)
tan(2x) / (1 + tan²(x))
sin²(x)
The double-angle formula for sine is sin(2x) = 2 sin(x) cos(x). The others are formulas for cosine or tangent. More at Khan Academy: Trig Identities.
Simplify sin(45° + ?).
(?2/2)(cos(?) + sin(?))
(?2/2)(cos(?) - sin(?))
sin(?) + cos(?)
?2 · sin(? + 45°)
Using the addition formula: sin(A + B) = sin A cos B + cos A sin B. Here A = 45°, B = ?, so sin45° = cos45° = ?2/2. More at Math is Fun: Trig Addition.
Solve for x in [0, 2?): 2 sin(x) - 1 = 0.
x = ?/6 and 5?/6
x = ?/3 and 2?/3
x = 7?/6 and 11?/6
x = ?/2 and 3?/2
Rearrange: sin(x) = 1/2. Solutions on [0, 2?) are x = ?/6 and 5?/6. Other angles give different sine values. See at Khan Academy: Trig Equations.
Find the inverse of f(x) = (3x - 4) / (5x + 2).
(4 + 2x) / (3 - 5x)
-(2x + 4) / (5x - 3)
(3x + 2) / (5 - 4x)
(2x - 4) / (5x + 3)
Swap x and y and solve: x = (3y - 4)/(5y + 2) ? y = -(2x + 4)/(5x - 3). This is the inverse. Detailed steps at Purplemath: Rational Inverse.
Solve for x: log?(x - 1) + log?(x + 1) = 3.
3
-3
±3
No solution
Combine logs: log?[(x - 1)(x + 1)] = 3 ? x² - 1 = 8 ? x² = 9. Domain requires x > 1, so x = 3 only. More at Math is Fun: Log Equations.
Which identity expresses tan(3x)?
(tan³ x + 3 tan x) / (3 tan² x - 1)
(3 tan x - tan³ x) / (1 - 3 tan² x)
(3 tan x + tan³ x) / (1 + 3 tan² x)
(tan³ x - 3 tan x) / (3 tan² x + 1)
Using angle addition twice: tan(2x + x) yields (3 tan x - tan³ x)/(1 - 3 tan² x). See derivation at Khan Academy: Triple-Angle Identities.
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Study Outcomes

  1. Understand Core Concepts -

    Grasp the foundational principles from Units 3, 4, and 5 by working through targeted questions that reinforce each section's key ideas.

  2. Apply Problem-Solving Skills -

    Use your knowledge to tackle unit 3 review quiz items and unit 5 review questions, strengthening your ability to solve representative problems.

  3. Identify Knowledge Gaps -

    Pinpoint areas of uncertainty through the unit 4 practice test and quiz feedback, helping you focus your revision where it matters most.

  4. Analyze Performance Trends -

    Evaluate your quiz results to see which topics you've mastered and which need more review, enabling a data-driven study approach.

  5. Boost Exam Confidence -

    Build assurance in your understanding and readiness by successfully completing a comprehensive unit trivia challenge that mirrors real exam conditions.

Cheat Sheet

  1. Quadratic Formula Mastery (Unit 3) -

    Review the quadratic formula x = [ - b ± √(b² - 4ac)]/(2a), as many unit 3 review quiz questions test your ability to identify a, b, and c quickly. Remember the discriminant mnemonic "D determines destiny": D>0 two real roots, D=0 one real root, D<0 two complex roots. Practice with 10 - 12 problems from MIT OpenCourseWare to solidify your speed and accuracy.

  2. Function Transformations (Unit 4) -

    Understand how f(x - h)+k shifts graphs: h moves horizontally, k moves vertically, which often appears on unit 4 practice test sections. A useful trick is "H before V" to apply horizontal translations first. Experiment with transformations on Desmos or Wolfram Alpha to see live updates and reinforce your intuition.

  3. Key Trigonometric Identities (Unit 4) -

    Memorize the Pythagorean identity sin²θ + cos²θ = 1 and the angle”sum formulas sin(α±β) and cos(α±β), as these form the backbone of many unit 4 review quiz problems. Use the mnemonic "Oscar Had A Heap Of Apples" for the unit circle quadrants (All, Sine, Tangent, Cosine). Solid practice from the University of Cambridge's trig worksheets will boost recall under timed conditions.

  4. Exponential & Logarithmic Rules (Unit 5) -

    Master the laws: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), and log_b(xy) = log_b x + log_b y, which are staples of any unit 5 review questions. Remember the change-of-base formula log_b a = log_c a / log_c b for effortless conversions on calculators. Khan Academy's interactive drills can help you internalize these rules in just 15 minutes a day.

  5. Derivative Applications & Optimization (Unit 5) -

    Practice the product, quotient, and chain rules alongside common optimization setups - maximizing area or minimizing cost - frequently featured in unit 5 review questions. A quick mnemonic for the product rule is "low·dhigh + high·dlow." Solve real-world problems from the American Mathematical Society's library to build confidence for exam-style prompts.

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