πŸ“
Self-Study Worksheet
Algebra 2
& Geometry
Core Problems Β· Memory Keys Β· Worked Examples
Name: _________________ Date: ___________ Period: ___
πŸ“˜ Algebra 2
πŸ“ Geometry
πŸ“˜ Algebra 2 β€” Key Problems
Q1 Β· Quadratic Formula β˜…β˜†β˜†βš  TRAP: sign of b
bΒ² βˆ’ 4ac is your FRIEND (discriminant = nature of roots)
✏ Example Solve \(x^2 - 5x + 6 = 0\)
β†’ Use \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\) with \(a=1,\, b=-5,\, c=6\)
β†’ \(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5\pm1}{2}\) β†’ \(x=3\) or \(x=2\) βœ“
Solve:  \(2x^2 + 3x - 5 = 0\)
Hint: identify \(a, b, c\) first. Don't forget the Β± !
Q2 Β· Vertex Form & Parabola β˜…β˜…β˜†
y = a(xβˆ’h)Β² + k β†’ vertex is (h, k). Watch the MINUS sign on h!
✏ Example \(y = 2(x-3)^2 + 1\) β†’ vertex \((3, 1)\), opens UP (a>0)
Convert to vertex form and find the vertex:
\(y = x^2 - 6x + 11\)
Step: complete the square. Add and subtract (b/2)Β²
Q3 Β· Systems of Equations β˜…β˜…β˜†βš  TRAP: substitution error
SUB-ELIM: Substitution or Elimination β€” pick the EASIER one!
✏ Example \(\begin{cases} y = 2x+1 \\ 3x + y = 16 \end{cases}\)
β†’ Sub \(y\): \(3x+(2x+1)=16\) β†’ \(5x=15\) β†’ \(x=3,\, y=7\)
Solve the system: \[\begin{cases} 2x - y = 4 \\ x + 3y = -5 \end{cases}\]
Q4 Β· Polynomial Division / Remainder Theorem β˜…β˜…β˜†
Remainder Theorem: plug in x=c directly into f(x) β†’ that IS the remainder!
✏ Example \(f(x)=x^3-4x+2\) divided by \((x-2)\)
Remainder = \(f(2)=8-8+2=2\) ← no long division needed! πŸŽ‰
Find the remainder when \(f(x) = 2x^3 - x^2 + 3x - 5\) is divided by \((x - 2)\).
A)  13
B)  17
C)  9
D)  21
Q5 Β· Exponential & Logarithms β˜…β˜…β˜…βš  TRAP: log(aΒ·b) β‰  log(a)Β·log(b)
log rules: PRODUCT β†’ ADD, QUOTIENT β†’ SUBTRACT, POWER β†’ MULTIPLY in front
✏ Example \(\log_2 8 = 3\) because \(2^3 = 8\)
\(\log(xy) = \log x + \log y\)  |  \(\log(x^n) = n\log x\)
Expand: \(\log_3\!\left(\dfrac{x^2 \sqrt{y}}{z}\right)\)
Use all three log rules. Show each step.
Also solve for \(x\):  \(3^{x+1} = 81\)
Q6 Β· Rational Expressions β˜…β˜…β˜†βš  TRAP: domain restriction β€” exclude values that make denominator = 0
FACTOR first, then CANCEL. Always state excluded values!
✏ Example \(\dfrac{x^2-4}{x+2} = \dfrac{(x+2)(x-2)}{x+2} = x-2, \quad x \neq -2\)
Simplify: \(\dfrac{x^2 - x - 6}{x^2 - 9}\)
Q7 Β· Arithmetic & Geometric Sequences β˜…β˜…β˜†
Arithmetic: ADD same d each time β†’ aβ‚™ = a₁ + (nβˆ’1)d
Geometric: MULTIPLY same r each time β†’ aβ‚™ = a₁ Β· rⁿ⁻¹
The 3rd term of a geometric sequence is 12 and the 6th term is 96. Find the common ratio \(r\) and the first term \(a_1\).
Hint: \(\dfrac{a_6}{a_3} = r^3\)
Q8 Β· Complex Numbers β˜…β˜…β˜†βš  TRAP: iΒ² = βˆ’1 (NOT +1!)
i cycle: iΒΉ=i, iΒ²=βˆ’1, iΒ³=βˆ’i, i⁴=1 β†’ REPEAT every 4!
✏ Example \((3+2i)(1-i) = 3-3i+2i-2i^2 = 3-i+2 = 5-i\)
Simplify: \((4 + 3i)^2\) and also find \(i^{47}\).
Q9 Β· Inverse Functions β˜…β˜…β˜…βš  TRAP: swap x and y BEFORE solving!
To find f⁻¹(x): Step 1 replace f(x) with y β†’ Step 2 SWAP x,y β†’ Step 3 solve for y
Find the inverse of \(f(x) = \dfrac{2x + 3}{x - 1}\)
Show all 3 steps. What is the domain restriction?
Q10 Β· Graphing Transformations β˜…β˜…β˜†
INSIDE parentheses β†’ horizontal (do OPPOSITE). OUTSIDE β†’ vertical (as written).
Negative in front β†’ REFLECT. Number in front β†’ STRETCH or SHRINK.
✏ Example \(f(x) = \sqrt{x}\) β†’ \(g(x) = -2\sqrt{x+3}-1\)
β†’ Left 3, reflect over x-axis, vertical stretch Γ—2, down 1
Describe ALL transformations applied to \(f(x)=x^2\) to get \(g(x) = -\tfrac{1}{2}(x-4)^2 + 5\)
β‘  Shift: ___________
β‘‘ Reflect: _________
β‘’ Stretch/Shrink: ___
β‘£ Vertical shift: ___
Page 1 / 2  Β·  Algebra 2
πŸ“ Geometry β€” Key Problems
G1 Β· Angle Relationships / Parallel Lines β˜…β˜†β˜†βš  TRAP: co-interior angles ADD to 180Β°, NOT equal
Alt. interior = EQUAL | Corresponding = EQUAL | Co-interior (same-side) = SUPPLEMENTARY (180Β°)
[Parallel lines ℓ₁ βˆ₯ β„“β‚‚ cut by transversal t]
Draw it here:                                    
Lines \(m \parallel n\). A transversal crosses them. One angle is \(112Β°\). Find all 8 angles.
G2 Β· Triangle Angle Sum & Exterior Angle β˜…β˜†β˜†
Interior angles of ANY triangle = 180Β°. Exterior angle = sum of the TWO NON-ADJACENT interiors.
In \(\triangle ABC\), \(\angle A = 3x+10Β°\), \(\angle B = 2x-5Β°\), \(\angle C = x+15Β°\). Find all angles.
G3 Β· Pythagorean Theorem & Special Right Triangles β˜…β˜…β˜†βš  TRAP: 45-45-90 legs are equal; 30-60-90 sides are x, x√3, 2x
45-45-90: legs = x β†’ hyp = x√2  |  30-60-90: short = x, long = x√3, hyp = 2x
✏ Example 30-60-90 with hyp = 10 β†’ short leg = 5, long leg = \(5\sqrt{3}\)
A 45-45-90 triangle has a hypotenuse of \(8\sqrt{2}\). Find the leg length.
Also: A 30-60-90 triangle has short leg = 7. Find the other two sides.
G4 Β· Triangle Congruence (SSS, SAS, ASA, AAS, HL) β˜…β˜…β˜†βš  TRAP: SSA is NOT a valid congruence! (Ambiguous case)
Valid: SSS, SAS, ASA, AAS, HL(right β–³)   INVALID: SSA, AAA (similarity only!)
State the congruence theorem that proves \(\triangle PQR \cong \triangle XYZ\):
a) \(PQ=XY,\, QR=YZ,\, PR=XZ\) β†’ ___
b) \(\angle P=\angle X,\, PQ=XY,\, \angle Q=\angle Y\) β†’ ___
c) \(PQ=XY,\, \angle Q=\angle Y,\, QR=YZ\) β†’ ___
d) \(\angle P=\angle X,\, \angle Q=\angle Y,\, PQ=XY\) β†’ ___
G5 Β· Similarity & Scale Factor β˜…β˜…β˜†
Similar β–³: sides PROPORTIONAL, angles EQUAL. Scale factor k β†’ area changes by kΒ²!
\(\triangle ABC \sim \triangle DEF\). \(AB = 9,\, BC = 12,\, DE = 6\). Find \(EF\) and the ratio of their areas.
Set up a proportion. Then square the scale factor for the area ratio.
G6 Β· Circle β€” Arcs, Chords & Inscribed Angles β˜…β˜…β˜…βš  TRAP: Inscribed angle = HALF the intercepted arc (not equal!)
Inscribed angle = Β½ Γ— intercepted arc  |  Central angle = intercepted arc (same!)
✏ Example Arc \(BC = 80Β°\) β†’ Inscribed \(\angle BAC = 40Β°\) (half of arc)
In circle \(O\), arc \(PQ = 130Β°\). Find the inscribed angle \(\angle PRQ\) where \(R\) is on the major arc.
Also: If a central angle is \(94Β°\), what is the inscribed angle that intercepts the same arc?
G7 Β· Area & Perimeter β€” Composite Figures β˜…β˜…β˜†
Composite = BREAK into simple shapes β†’ ADD or SUBTRACT areas. Draw the cuts!
A rectangle \(10 \times 8\) has a semicircle of diameter 6 cut out of one end. Find the area of the remaining figure. (Use \(\pi \approx 3.14\))
G8 Β· Volume & Surface Area β€” 3D Solids β˜…β˜…β˜…βš  TRAP: Cone/Pyramid use β…“ in front of formula!
Cylinder: V=Ο€rΒ²h  |  Cone: V=β…“Ο€rΒ²h  |  Sphere: V=⁴⁄₃πrΒ³  |  Prism: V=Bh
A cone has radius 5 and height 12. Find the volume AND lateral surface area.
(Slant height: \(\ell = \sqrt{r^2 + h^2}\),   Lateral SA = \(\pi r \ell\))
G9 Β· Coordinate Geometry β€” Distance, Midpoint, Slope β˜…β˜…β˜†
Distance = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)  |  Midpoint = \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)
Parallel lines: same slope. Perpendicular: slopes are NEGATIVE RECIPROCALS (product = βˆ’1)
Points \(A(-2,\,3)\) and \(B(4,\,-1)\).
(a) Find the distance \(AB\).
(b) Find the midpoint \(M\) of \(\overline{AB}\).
(c) A line perpendicular to \(AB\) passes through \(M\). Write its equation.
G10 Β· Trigonometry β€” SOH-CAH-TOA & Law of Sines/Cosines β˜…β˜…β˜…βš  TRAP: sin⁻¹ is NOT 1/sin ! It means "inverse sine"
SOH: sin=Opp/Hyp  |  CAH: cos=Adj/Hyp  |  TOA: tan=Opp/Adj
Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}\)  |  Law of Cosines: \(c^2=a^2+b^2-2ab\cos C\)
✏ Example Right β–³, angle 35Β°, hypotenuse = 10 β†’ opposite = \(10\sin 35Β° \approx 5.74\)
In \(\triangle ABC\): \(a = 7,\, b = 9,\, C = 55Β°\). Find side \(c\) using the Law of Cosines.
Also: A right triangle has legs 5 and 12. Find all angles.
Page 2 / 2  Β·  Geometry