Unit 1–10
Algebra 2
Algebra 2
Core Problems
Key topics · Tricky questions · Memory points included
Progress
0 / 10 answered
⚡ Memory Point
VERTEX = opposite sign of h, same sign of k · y = a(x−h)² + k
Which of the following is the vertex of the parabola \( y = 2(x - 3)^2 + 5 \)?
✓ Correct! Explanation
In vertex form \(y = a(x-h)^2 + k\), the vertex is \((h, k)\). Here \(h = 3\) and \(k = 5\), so vertex = \((3, 5)\). Common mistake: students flip the sign of \(h\) because of the minus sign — remember \(x - 3 = 0\) gives \(x = 3\).
⚡ Memory Point
DISCRIMINANT tells the number of solutions: \(b^2 - 4ac\) · + = 2 real, 0 = 1 real, − = 2 imaginary
How many real solutions does \( 3x^2 - 6x + 4 = 0 \) have?
Hint: calculate b² − 4ac first → 36 − 48 = ?
✓ Correct! Explanation
\(b^2 - 4ac = (-6)^2 - 4(3)(4) = 36 - 48 = -12 < 0\). A negative discriminant means NO real solutions — the parabola never crosses the x-axis. The solutions are complex (imaginary).
⚡ Memory Point
END BEHAVIOR: even degree → same direction both ends · odd degree → opposite directions
What is the end behavior of \( f(x) = -2x^4 + 3x^2 - 1 \)?
✓ Correct! Explanation
Degree 4 is even, leading coefficient is −2 (negative). Even + negative → both ends go DOWN to \(-\infty\). Think of \(-x^4\) which is an upside-down U shape — both arms point down. Answer: B.
⚡ Memory Point
POWER/ROOT: \(x^{m/n} = \sqrt[n]{x^m}\) · denominator = index of root, numerator = power
Simplify: \( 27^{2/3} \)
✓ Correct! Explanation
\(27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9\). Step 1: cube root of 27 = 3. Step 2: square it = 9. Always take the root FIRST to keep numbers small. Answer: C.
⚡ Memory Point
LOG ↔ EXPONENT: \(\log_b x = y\) means \(b^y = x\) · "base to what power = argument?"
Solve for \(x\): \( \log_3(x) = 4 \)
✓ Correct! Explanation
Convert to exponential form: \(\log_3(x) = 4 \Rightarrow 3^4 = x \Rightarrow x = 81\). The base (3) raised to the log value (4) equals the argument (x). Answer: C.
⚡ Memory Point
GROWTH vs DECAY: \(y = a \cdot b^x\) — if \(b > 1\) → growth, if \(0 < b < 1\) → decay
A population starts at 500 and doubles every year. Which equation models the population \(P\) after \(t\) years?
✓ Correct! Explanation
Exponential growth: initial value × (growth factor)^time. Here: \(P = 500 \cdot 2^t\). After 1 year: 500·2=1000. After 2 years: 500·4=2000. Never add for doubling — always multiply. Answer: B.
⚡ Memory Point
SUBSTITUTION: isolate one variable → plug in → solve → back-substitute to find the other
Solve the system: \( y = 2x + 1 \) and \( 3x + y = 16 \)
✓ Correct! Explanation
Substitute \(y = 2x+1\) into \(3x + y = 16\): \(3x + (2x+1) = 16 \Rightarrow 5x = 15 \Rightarrow x = 3\). Then \(y = 2(3)+1 = 7\). Answer: A — \((3, 7)\). Always verify both equations!
⚡ Memory Point
\(i^2 = -1\) · Cycle: \(i^1=i,\; i^2=-1,\; i^3=-i,\; i^4=1\) then repeats!
Simplify: \( (3 + 2i)(1 - 4i) \)
✓ Correct! Explanation
Use FOIL: \((3)(1) + (3)(-4i) + (2i)(1) + (2i)(-4i)\) \(= 3 - 12i + 2i - 8i^2\) \(= 3 - 10i - 8(-1)\) \(= 3 - 10i + 8 = 11 - 10i\). Key: \(i^2 = -1\) turns into a REAL number! Answer: B.
⚡ Memory Point
GEOMETRIC sum: \(S_n = \dfrac{a_1(1 - r^n)}{1 - r}\) · r = common ratio (multiply each time)
Find the sum of the first 4 terms of the geometric sequence: \(2,\; 6,\; 18,\; 54,\;\ldots\)
✓ Correct! Explanation
Just add the 4 terms: \(2 + 6 + 18 + 54 = 80\). Or use formula: \(S_4 = \frac{2(1-3^4)}{1-3} = \frac{2(-80)}{-2} = 80\). Common ratio \(r = 3\). Answer: B.
⚡ Memory Point
ELLIPSE: \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) — bigger denominator = longer axis direction!
Which is the equation of an ellipse with a horizontal major axis of length 10 and a vertical minor axis of length 6, centered at the origin?
✓ Correct! Explanation
Major axis length 10 → \(a = 5\) → \(a^2 = 25\) (under x² because horizontal). Minor axis length 6 → \(b = 3\) → \(b^2 = 9\) (under y²). Answer: A. Trap: don't use the full length (10, 6) — use HALF (semi-axis: 5, 3).
Unit 1–10
Geometry
Geometry
Core Problems
Key topics · Tricky questions · Memory points included
Progress
0 / 10 answered
⚡ Memory Point
PARALLEL LINES cut by transversal: alternate interior = equal · co-interior (same-side) = 180°
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \(72°\). What is the other co-interior angle?
✓ Correct! Explanation
Co-interior (consecutive interior) angles are supplementary: they add to 180°. So \(180° - 72° = 108°\). Don't confuse with alternate interior angles (which are equal). Answer: B.
⚡ Memory Point
CONGRUENCE shortcuts: SSS, SAS, ASA, AAS, HL(right triangles only) · SSA is NOT valid!
Two triangles have two pairs of equal sides and the angle between those sides equal. Which congruence rule applies?
✓ Correct! Explanation
Two sides AND the INCLUDED angle (between the two sides) = SAS (Side-Angle-Side). The key word is "between" — the angle must be sandwiched between the two sides. Answer: C.
⚡ Memory Point
\(a^2 + b^2 = c^2\) — c is ALWAYS the hypotenuse (longest side, opposite right angle)
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
✓ Correct! Explanation
\(c^2 = 5^2 + 12^2 = 25 + 144 = 169 \Rightarrow c = 13\). The 5-12-13 is a Pythagorean triple — memorize it! Common mistake: choosing 169 (that's \(c^2\), not \(c\)). Answer: A.
⚡ Memory Point
INSCRIBED ANGLE = HALF the intercepted arc · Central angle = full arc
An inscribed angle intercepts an arc of \(140°\). What is the measure of the inscribed angle?
✓ Correct! Explanation
Inscribed Angle Theorem: inscribed angle = ½ × intercepted arc. \(\frac{1}{2} \times 140° = 70°\). The most common mistake is choosing 140° (confusing inscribed with central angle). Answer: B.
⚡ Memory Point
TRAPEZOID area: \(\dfrac{(b_1 + b_2)}{2} \times h\) · average the TWO bases, then multiply by height
Find the area of a trapezoid with bases of 8 cm and 14 cm, and a height of 5 cm.
✓ Correct! Explanation
\(A = \frac{(8 + 14)}{2} \times 5 = \frac{22}{2} \times 5 = 11 \times 5 = 55 \text{ cm}^2\). Answer: A. Trap: students often forget to divide by 2, getting 110.
⚡ Memory Point
SIMILAR = same shape, different size · corresponding sides are PROPORTIONAL (set up a ratio equation)
Two similar triangles have corresponding sides in ratio 3:5. If the smaller triangle has a perimeter of 24, what is the perimeter of the larger triangle?
✓ Correct! Explanation
Perimeters are in the SAME ratio as sides: \(\frac{3}{5} = \frac{24}{x} \Rightarrow x = \frac{24 \times 5}{3} = 40\). Answer: B. Note: for AREAS the ratio would be \(3^2 : 5^2 = 9:25\).
⚡ Memory Point
CYLINDER: Volume = \(\pi r^2 h\) · Surface Area = \(2\pi r^2 + 2\pi r h\) · (2 circles + rectangle wrapped around)
Find the volume of a cylinder with radius 3 cm and height 10 cm. (Use \(\pi \approx 3.14\))
✓ Correct! Explanation
\(V = \pi r^2 h = 3.14 \times 3^2 \times 10 = 3.14 \times 9 \times 10 = 282.6 \text{ cm}^3\). Common error: using diameter (6) instead of radius (3). Always halve the diameter first! Answer: B.
⚡ Memory Point
MIDPOINT: average the x's, average the y's → \(\left(\dfrac{x_1+x_2}{2},\; \dfrac{y_1+y_2}{2}\right)\)
What is the midpoint of the segment with endpoints \((−4,\; 6)\) and \((10,\; −2)\)?
✓ Correct! Explanation
\(x = \frac{-4+10}{2} = \frac{6}{2} = 3\), \(y = \frac{6+(-2)}{2} = \frac{4}{2} = 2\). Midpoint = \((3, 2)\). Watch the negatives carefully! Answer: A.
⚡ Memory Point
REFLECTION over x-axis: \((x, y) \to (x, -y)\) · over y-axis: \((x, y) \to (-x, y)\)
Point \(P = (4, -3)\) is reflected over the y-axis. What are the coordinates of \(P'\)?
✓ Correct! Explanation
Reflection over y-axis flips the x-coordinate: \((x,y) \to (-x, y)\). So \((4, -3) \to (-4, -3)\). The y-coordinate stays the same. Answer: C. Trap: students often change both coordinates.
⚡ Memory Point
SOH-CAH-TOA: sin = Opp/Hyp · cos = Adj/Hyp · tan = Opp/Adj · always relative to the GIVEN angle
In a right triangle, the angle \(\theta = 30°\), and the hypotenuse is 20. What is the length of the side opposite to \(\theta\)?
✓ Correct! Explanation
Use SOH: \(\sin(30°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{20}\). Since \(\sin(30°) = \frac{1}{2}\): \(x = 20 \times \frac{1}{2} = 10\). Memorize: sin 30° = 0.5, sin 45° ≈ 0.707, sin 60° ≈ 0.866. Answer: A.