Algebra 2 & Geometry Practice

Q1 Quadratics

VERTEX = ( −b/2a , f(−b/2a) ) · MAX/MIN depends on sign of "a"

A ball is launched upward. Its height (in feet) after $t$ seconds is given by:

$$h(t) = -16t^2 + 64t + 5$$

What is the maximum height the ball reaches?

📖 Step-by-step: Maximum height occurs at the vertex. Find $t = -b/2a = -64/(2 \times -16) = 2$ seconds.
Then $h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = \mathbf{69}$ feet.
Trap: Many students forget to plug $t$ back in and stop at $t = 2$.

Q2 Systems of Equations

SUBSTITUTION: isolate one variable first · CHECK both equations after solving

Two car rental companies charge the following:

Company A: $\$30$ flat fee + $\$0.25$ per mileCompany B: $\$10$ flat fee + $\$0.45$ per mile

At how many miles do both companies cost the same?

📖 Step-by-step: Set equal: $30 + 0.25m = 10 + 0.45m$
$20 = 0.20m \Rightarrow m = \mathbf{100}$ miles.
Cost at 100 miles = $30 + 25 = \$55$ for both. ✓

Q3 Exponential Growth

GROWTH: A = P(1 + r)ᵗ · DECAY: A = P(1 − r)ᵗ · r is a DECIMAL

A bacteria culture starts with 500 cells and doubles every 3 hours.

Which expression gives the number of cells after $t$ hours?

📖 Key insight: "Doubles every 3 hours" → base is 2, but exponent must be $t/3$ so the doubling happens every 3 hours, not every 1 hour.
At $t=3$: $500 \cdot 2^1 = 1000$ ✓ At $t=6$: $500 \cdot 2^2 = 2000$ ✓

Q4 Logarithms

log = EXPONENT · log_b(x) = y means b^y = x · "Change of Base": log_b(x) = ln(x)/ln(b)

An investment grows according to $A = 1000 \cdot e^{0.05t}$.

Approximately how many years does it take for the investment to reach $\$2000$?

📖 Step-by-step: $2000 = 1000 \cdot e^{0.05t}$ → $2 = e^{0.05t}$ → $\ln 2 = 0.05t$ → $t = \ln 2 / 0.05 \approx 0.693/0.05 \approx \mathbf{13.86}$ years.

Q5 Polynomials

REMAINDER THEOREM: remainder when P(x) ÷ (x−a) = P(a) · FACTOR if P(a)=0

The polynomial $P(x) = x^3 - 4x^2 + x + 6$ is divided by $(x - 2)$.

What is the remainder?

📖 Remainder Theorem: Substitute $x = 2$: $P(2) = 8 - 16 + 2 + 6 = \mathbf{0}$.
Since remainder = 0, $(x-2)$ is a factor of $P(x)$.

Q6 Rational Functions

VERTICAL ASYMPTOTE: denominator = 0 · HORIZONTAL: compare degrees of top vs bottom

A factory produces $x$ units at a total cost of $C(x) = \frac{500x + 2000}{x}$ dollars per unit.

As production increases indefinitely, what does the cost per unit approach?

📖 Horizontal Asymptote: $C(x) = 500 + 2000/x$. As $x \to \infty$, $2000/x \to 0$, so $C(x) \to \mathbf{500}$.
This is the horizontal asymptote: $y = 500$.

Q7 Complex Numbers

i² = −1 · i³ = −i · i⁴ = 1 · CYCLE repeats every 4

A physics model involves the expression $(3 + 2i)(1 - 4i)$.

Expand and simplify. What is the result?

📖 FOIL with i: $(3)(1) + (3)(-4i) + (2i)(1) + (2i)(-4i)$
$= 3 - 12i + 2i - 8i^2$
$= 3 - 10i - 8(-1) = 3 - 10i + 8 = \mathbf{11 - 10i}$
Trap: forgetting $i^2 = -1$ turns the $-8i^2$ into $-8$ not $+8$.

Q8 Sequences & Series

GEOMETRIC SUM (finite): S = a(1−rⁿ)/(1−r) · INFINITE (|r|<1): S = a/(1−r)

A bouncing ball drops from 10 ft and each bounce reaches 60% of the previous height.

What is the total vertical distance traveled (down + up for all bounces), starting from the first drop? (Use the infinite series model.)

📖 Two-phase approach: Initial fall = $10$ ft.
Each bounce: ball goes up then down. First bounce up = $10 \times 0.6 = 6$ ft. Sum of all bounce distances = $2 \times \frac{6}{1-0.6} = 2 \times 15 = 30$ ft.
Total = $10 + 30 = \mathbf{40}$ ft.

Q9 Matrices

MATRIX MULTIPLY: (m×n)(n×p) = (m×p) · Row × Column · ORDER MATTERS

A store sells apples at $\$1.20$ and bananas at $\$0.50$ each. On Monday it sold 30 apples and 20 bananas; on Tuesday, 15 apples and 40 bananas.

This can be modeled as a matrix product. What was the total revenue on Tuesday?

📖 Step-by-step: Tuesday: $15 \times \$1.20 + 40 \times \$0.50 = \$18 + \$20 = \mathbf{\$38}$.
Matrix form: $[15 \; 40] \cdot \begin{bmatrix}1.20\\0.50\end{bmatrix} = 38$

Q10 Radical & Rational Exponents

x^(a/b) = (ᵇ√x)ᵃ · ALWAYS check for EXTRANEOUS solutions when squaring both sides

Solve: $\sqrt{3x + 4} = x - 2$

📖 Check extraneous solutions! Square both sides: $3x+4 = x^2-4x+4$ → $x^2-7x=0$ → $x(x-7)=0$ → $x=0$ or $x=7$.
Check $x=0$: $\sqrt{4} = -2$? → $2 \neq -2$ ✗ Extraneous!
Check $x=7$: $\sqrt{25} = 5$? → $5 = 5$ ✓
Answer: $\mathbf{x = 7}$ only.

Q11 Pythagorean Theorem

a² + b² = c² · c is ALWAYS the HYPOTENUSE (longest side, opposite right angle)

A 26-foot ladder leans against a wall. Its base is 10 feet from the wall.

How high up the wall does the ladder reach?

📖 Pythagorean Theorem: $a^2 + 10^2 = 26^2$ → $a^2 = 676 - 100 = 576$ → $a = \mathbf{24}$ feet.
Tip: Recognize common triples: 10-24-26 is 5-12-13 scaled by 2.

Q12 Circles

INSCRIBED ANGLE = half of CENTRAL angle · Both intercept the same arc

In circle O, arc $AB = 140°$. Chord $CD$ is parallel to $AB$, and arc $CD = 80°$.

An inscribed angle intercepts arc $AB$. What is the measure of that inscribed angle?

📖 Inscribed Angle Theorem: Inscribed angle = $\frac{1}{2}$ × intercepted arc = $\frac{1}{2} \times 140° = \mathbf{70°}$.
Central angle = arc (same). Inscribed angle = half the arc.

Q13 Similar Triangles

AA · SAS · SSS similarity · CORRESPONDING sides are PROPORTIONAL

A 6-ft person casts a 4-ft shadow at the same time a nearby flagpole casts a 22-ft shadow.

How tall is the flagpole?

📖 Proportion setup: $\frac{6}{4} = \frac{h}{22}$ → $h = \frac{6 \times 22}{4} = \frac{132}{4} = \mathbf{33}$ ft.

Q14 Coordinate Geometry

MIDPOINT: ((x₁+x₂)/2, (y₁+y₂)/2) · DISTANCE: √[(Δx)²+(Δy)²]

Point $M(3, -1)$ is the midpoint of segment $\overline{AB}$ where $A = (1, 5)$.

Find the coordinates of point $B$.

📖 Work backwards from midpoint: $\frac{1 + x_B}{2} = 3 \Rightarrow x_B = 5$
$\frac{5 + y_B}{2} = -1 \Rightarrow y_B = -7$
$B = \mathbf{(5, -7)}$

Q15 Surface Area & Volume

CYLINDER: V = πr²h · SA = 2πr² + 2πrh · CONE: V = ⅓πr²h

A cylindrical water tank has a radius of 5 m and a height of 12 m.

How much water (in m³) does it hold? Use $\pi \approx 3.14$.

📖 Cylinder Volume: $V = \pi r^2 h = 3.14 \times 25 \times 12 = 3.14 \times 300 = \mathbf{942}$ m³.
Trap: Using diameter (10) instead of radius (5) gives $r^2 = 100$, quadrupling the answer.

Q16 Trigonometry (Right Triangles)

SOH-CAH-TOA · sin=O/H · cos=A/H · tan=O/A · "Oscar Has A Hat On Always"

From a point 50 m from the base of a building, the angle of elevation to the top is 62°.

How tall is the building? (Use $\tan 62° \approx 1.88$)

📖 tan(angle) = opposite / adjacent: $\tan 62° = h/50$ → $h = 50 \times 1.88 = \mathbf{94}$ m.
The 50 m is the ADJACENT side (horizontal ground distance), h is OPPOSITE (vertical height).

Q17 Triangle Congruence

SSS · SAS · ASA · AAS · HL (right triangles only) · NOT SSA (the "ambiguous case")

Two triangles share a common side. You are told two angles and the included side of one equal two angles and the included side of the other.

Which congruence postulate applies?

📖 ASA vs AAS: ASA: angle–SIDE–angle (side is BETWEEN the two known angles).
AAS: angle–angle–SIDE (side is NOT between the angles).
Here the side is included (between the two angles) → ASA.

Q18 Parallel Lines & Transversals

ALTERNATE INTERIOR = equal · CO-INTERIOR (same-side) = 180° · CORRESPONDING = equal

Two parallel lines are cut by a transversal. One angle measures $(3x + 15)°$ and its co-interior (same-side interior) angle measures $(2x + 25)°$.

Find $x$.

📖 Co-interior angles are supplementary (sum = 180°): $(3x+15) + (2x+25) = 180$ → $5x + 40 = 180$ → $5x = 140$ → $x = \mathbf{28}$

Q19 Transformations

ROTATION 90° CCW: (x,y)→(−y,x) · 180°: (x,y)→(−x,−y) · REFLECTION over y-axis: (x,y)→(−x,y)

Point $P(4, -3)$ is rotated $90°$ counterclockwise about the origin.

What are the new coordinates?

📖 90° CCW rule: $(x, y) \to (-y, x)$
$P(4, -3) \to (- (-3), 4) = \mathbf{(3, 4)}$
Memory trick: "Swap and negate the first" for 90° CCW.

Q20 Area of Composite Figures

COMPOSITE = ADD simple shapes OR SUBTRACT · Draw it! Split into rectangles, triangles, circles

A running track consists of a rectangle (100 m × 60 m) with two semicircles on each short end.

What is the total area enclosed by the track? (Use $\pi \approx 3.14$)

📖 Rectangle + full circle (two semicircles = one full circle): Rectangle: $100 \times 60 = 6000$ m²
Full circle radius = $60/2 = 30$ m: $\pi r^2 = 3.14 \times 900 = 2826$ m²
Total = $6000 + 2826 = \mathbf{8826}$ m²

Core Problems
That Actually Trick You

Algebra 2

Geometry

You finished! 🎉

Core ProblemsThat Actually Trick You

Algebra 2

Geometry

You finished! 🎉

Core Problems
That Actually Trick You