Algebra 2 & Geometry — Study Guide

Q01 Algebra 2 Quadratics ★☆☆

KEY: "thrown upward" → vertex = max height, set h(t)=0 for landing

A ball is thrown upward from the ground with an initial velocity of 48 ft/s. Its height is modeled by \( h(t) = -16t^2 + 48t \). How long does it take to reach its maximum height?

Quick Example (Simpler) If \( h(t) = -16t^2 + 32t \), the vertex is at \( t = \frac{-32}{2(-16)} = 1 \) second. Use the same formula: \( t = \frac{-b}{2a} \).

Q02 Algebra 2 Systems ★★☆

KEY: "together" → add rates; "opposite directions" → add speeds

Pipe A fills a tank in 4 hours. Pipe B fills the same tank in 6 hours. If both pipes are open together, how many hours does it take to fill the tank?

Rate Rule Rate of A = 1/4 tank/hr. Rate of B = 1/6 tank/hr. Combined: \(\frac{1}{4} + \frac{1}{6} = \frac{5}{12}\). Time = \(\frac{1}{\frac{5}{12}} = \frac{12}{5}\).

Q03 Algebra 2 Exponentials ★☆☆

KEY: doubling → \(A = A_0 \cdot 2^{t/d}\) where d = doubling period

A bacteria culture starts with 500 cells and doubles every 3 hours. Which expression gives the number of cells after \(t\) hours?

Watch the trap! Many students write \(500 \cdot 2^t\) — forgetting to divide \(t\) by the doubling period. After 3 hours you should get \(500 \times 2^1 = 1000\), not \(500 \times 2^3 = 4000\).

Q04 Algebra 2 Logarithms ★★☆

KEY: log = exponent; "log base b of x = y" means b^y = x

An investment doubles when \(\log_2(A) = 5\). What is the value of \(A\)?

Convert log → exponential \(\log_2(A) = 5\) means \(2^5 = A\). The base goes to the bottom, the log value becomes the exponent.

Q05 Algebra 2 Rational Equations ★★☆

TRAP: always check extraneous solutions — plug back in!

Solve: \(\dfrac{x}{x-2} + \dfrac{3}{x+2} = \dfrac{8}{x^2-4}\)

Key insight Notice \(x^2 - 4 = (x-2)(x+2)\). Multiply both sides by this LCD to clear all fractions. Then solve — and verify \(x \ne 2\) and \(x \ne -2\).

Q06 Algebra 2 Complex Numbers ★★☆

KEY: i² = −1; treat like FOIL but replace i² with −1 at the end

Simplify: \((3 + 2i)(1 - 4i)\)

Q07 Algebra 2 Sequences ★☆☆

KEY: arithmetic → add d; geometric → multiply r. Don't mix them up!

The 3rd term of an arithmetic sequence is 11 and the 7th term is 27. What is the common difference \(d\)?

Strategy From term 3 to term 7 is 4 steps. Difference in value: 27 − 11 = 16. So \(d = \frac{16}{4} = 4\).

Q08 Algebra 2 Polynomials ★★★

KEY: Remainder Theorem — f(c) gives the remainder when divided by (x−c)

When \(f(x) = x^3 - 2x^2 + 5x - 6\) is divided by \((x-2)\), what is the remainder?

Q09 Algebra 2 Conic Sections ★★☆

KEY: circle = (x−h)²+(y−k)²=r²; center is (h, k) — watch the SIGNS!

What is the center of the circle \((x+3)^2 + (y-5)^2 = 49\)?

Sign Trap! \((x \mathbf{+} 3)^2\) means \(h = \mathbf{-3}\), not +3. The standard form is \((x-h)^2\), so the sign flips.

Q10 Algebra 2 Probability ★★☆

KEY: P(A or B) = P(A) + P(B) − P(A and B) — don't double-count!

In a class of 30, 18 play soccer, 12 play basketball, and 5 play both. What is the probability a randomly chosen student plays at least one sport?

Q11 Geometry Triangles ★☆☆

KEY: Pythagorean Theorem — a²+b²=c², c is always the HYPOTENUSE

A ladder 13 ft long leans against a wall. Its base is 5 ft from the wall. How high up the wall does it reach?

Visualize it The ladder = hypotenuse (c = 13). Base = one leg (a = 5). Wall height = other leg (b = ?). \(5^2 + b^2 = 13^2\) → \(b^2 = 169 - 25 = 144\) → \(b = 12\).

Q12 Geometry Circles ★☆☆

KEY: Arc length = (θ/360) × 2πr; Area of sector = (θ/360) × πr²

A circle has radius 10 cm. What is the arc length of a sector with a central angle of 72°?

Q13 Geometry Similar Triangles ★★☆

KEY: similar → ratios of corresponding sides are EQUAL (set up a proportion)

A 6 ft person casts a 4 ft shadow. At the same time, a tree casts a 22 ft shadow. How tall is the tree?

Q14 Geometry Volume ★★☆

KEY: cone volume = ⅓πr²h; sphere = (4/3)πr³ — the ⅓ is the most-forgotten part

A cone has a radius of 3 cm and a height of 7 cm. What is its exact volume?

Q15 Geometry Angles ★☆☆

KEY: parallel lines + transversal → alternate interior angles are EQUAL

Two parallel lines are cut by a transversal. One angle formed is \(3x + 15\)° and its alternate interior angle is \(5x - 25\)°. Find \(x\).

Q16 Geometry Surface Area ★★☆

KEY: cylinder SA = 2πr² + 2πrh (two circles + rectangle wrapped around)

A cylindrical can has radius 4 cm and height 10 cm. What is its total surface area?

Q17 Geometry Coordinate Geometry ★☆☆

KEY: midpoint = average the x's, average the y's: \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)

Segment \(\overline{AB}\) has endpoints \(A(-3, 4)\) and \(B(7, -2)\). What is the midpoint?

Q18 Geometry Polygons ★★☆

KEY: sum of interior angles = (n−2)×180°; one angle (regular) = that ÷ n

What is the measure of each interior angle of a regular octagon?

Quick Check An octagon has 8 sides. Sum = (8−2)×180 = 6×180 = 1080°. Each angle = 1080÷8 = 135°.

Q19 Geometry Trigonometry ★★☆

KEY: SOH-CAH-TOA · sin=Opp/Hyp · cos=Adj/Hyp · tan=Opp/Adj

In right triangle ABC, angle A = 30°, and the hypotenuse AC = 20. What is the length of the side opposite angle A (side BC)?

SOH! sin(A) = Opposite/Hypotenuse → sin(30°) = BC/20 → 0.5 = BC/20 → BC = 10.

Q20 Geometry Proofs / Congruence ★★★

KEY: SAS=Side-Angle-Side · ASA=Angle-Side-Angle · SSS · AAS · HL(right Δ only)

Two triangles share a common side. You know two pairs of sides are equal, and the included angle (between those sides) is equal. Which congruence postulate applies?

The Trap SSA is NOT a valid congruence rule (it's sometimes called the "ambiguous case"). The angle must be INCLUDED (between the two known sides) for SAS to work.