Algebra 2
Quadratics · Functions · Exponents · Logarithms · Sequences
⚡ DISCRIMINANT RULE: \(b^2 - 4ac\) → >0 two real, =0 one real, <0 no real
Quick Example
For \(x^2 - 5x + 6 = 0\): discriminant = \(25 - 24 = 1 > 0\) → two real roots ✓
How many real solutions does \(2x^2 - 4x + 5 = 0\) have?
Hint: Compute the discriminant \(\Delta = b^2 - 4ac\) first.
📖 해설
\(a=2,\ b=-4,\ c=5\)
판별식 \(\Delta = (-4)^2 - 4(2)(5) = 16 - 40 = -24\)
판별식이 음수이므로 실수 근이 없습니다. 정답은 C.
판별식 \(\Delta = (-4)^2 - 4(2)(5) = 16 - 40 = -24\)
판별식이 음수이므로 실수 근이 없습니다. 정답은 C.
⚡ VERTEX FORM: \(y = a(x-h)^2 + k\) → vertex is \((h,\ k)\)
Quick Example
\(y = 3(x-2)^2 + 7\) → vertex \((2, 7)\). Watch the sign: \((x\mathbf{-}2)\) means \(h = \mathbf{+}2\)!
What is the vertex of \(y = -2(x+3)^2 - 1\)?
📖 해설
\(y = -2(x - (-3))^2 + (-1)\) 형태이므로
\(h = -3,\ k = -1\) → 꼭짓점 \((-3,\ -1)\)
괄호 안이 \((x+3)\)이면 \(h = -3\)임에 주의! 정답은 B.
\(h = -3,\ k = -1\) → 꼭짓점 \((-3,\ -1)\)
괄호 안이 \((x+3)\)이면 \(h = -3\)임에 주의! 정답은 B.
⚡ NEGATIVE EXPONENT: \(a^{-n} = \dfrac{1}{a^n}\) · ZERO EXPONENT: \(a^0 = 1\)
Quick Example
\(3^{-2} = \dfrac{1}{9}\) and \(5^0 = 1\). These trip people up constantly!
Simplify: \(\dfrac{x^{-3} \cdot x^5}{x^{-1}}\)
📖 해설
지수 법칙: 분자 \(x^{-3+5} = x^2\), 이것을 \(x^{-1}\)로 나누면
\(x^{2-(-1)} = x^{2+1} = x^3\). 정답은 C.
\(x^{2-(-1)} = x^{2+1} = x^3\). 정답은 C.
⚡ LOG-EXP INVERSE: \(\log_b(b^x) = x\) and \(b^{\log_b x} = x\)
Quick Example
\(\log_2 8 = \log_2 2^3 = 3\). Ask yourself: "2 to what power gives 8?"
Evaluate: \(\log_3 \dfrac{1}{27}\)
📖 해설
\(\dfrac{1}{27} = 3^{-3}\)이므로
\(\log_3 3^{-3} = -3\). "3을 몇 제곱해야 \(\frac{1}{27}\)?" → \(-3\). 정답은 D.
\(\log_3 3^{-3} = -3\). "3을 몇 제곱해야 \(\frac{1}{27}\)?" → \(-3\). 정답은 D.
⚡ LOG PRODUCT RULE: \(\log(ab) = \log a + \log b\) · QUOTIENT: \(\log\!\frac{a}{b} = \log a - \log b\)
Given \(\log_5 2 = 0.431\) and \(\log_5 3 = 0.682\), find \(\log_5 6\).
📖 해설
\(6 = 2 \times 3\)이므로
\(\log_5 6 = \log_5 2 + \log_5 3 = 0.431 + 0.682 = 1.113\). 정답은 B.
\(\log_5 6 = \log_5 2 + \log_5 3 = 0.431 + 0.682 = 1.113\). 정답은 B.
⚡ nth TERM: \(a_n = a_1 + (n-1)d\) where \(d\) = common difference
Quick Example
Sequence \(3, 7, 11, \ldots\): \(d=4\), so \(a_{10} = 3 + 9(4) = 39\).
The arithmetic sequence begins \(5,\ 11,\ 17,\ \ldots\) What is the 20th term?
📖 해설
\(a_1 = 5,\ d = 6\)
\(a_{20} = 5 + (20-1)(6) = 5 + 114 = 119\). 정답은 C.
\(a_{20} = 5 + (20-1)(6) = 5 + 114 = 119\). 정답은 C.
⚡ GEOMETRIC nth TERM: \(a_n = a_1 \cdot r^{n-1}\) where \(r\) = common ratio
The geometric sequence \(4,\ 12,\ 36,\ \ldots\) What is the 6th term?
📖 해설
\(a_1 = 4,\ r = 3\)
\(a_6 = 4 \cdot 3^{6-1} = 4 \cdot 243 = 972\). 정답은 B.
\(a_6 = 4 \cdot 3^{6-1} = 4 \cdot 243 = 972\). 정답은 B.
⚡ MAX HEIGHT TIME: \(t = -\dfrac{b}{2a}\) from vertex formula
A ball is thrown upward. Its height (in meters) after \(t\) seconds is \(h(t) = -5t^2 + 20t + 3\). What is the maximum height?
📖 해설
꼭짓점의 \(t\) 좌표: \(t = -\dfrac{20}{2(-5)} = 2\)초
최대 높이: \(h(2) = -5(4) + 20(2) + 3 = -20 + 40 + 3 = 23\) m. 정답은 C.
최대 높이: \(h(2) = -5(4) + 20(2) + 3 = -20 + 40 + 3 = 23\) m. 정답은 C.
⚡ COMPOSITION ORDER: \((f \circ g)(x) = f(g(x))\) — plug \(g\) INTO \(f\), not the other way!
Quick Example
If \(f(x)=2x\) and \(g(x)=x+1\), then \(f(g(3)) = f(4) = 8\).
Let \(f(x) = x^2 + 1\) and \(g(x) = 3x - 2\). Find \(f(g(2))\).
📖 해설
먼저 \(g(2) = 3(2)-2 = 4\)
그다음 \(f(4) = 4^2 + 1 = 17\). 정답은 C.
그다음 \(f(4) = 4^2 + 1 = 17\). 정답은 C.
⚡ EXCLUDED VALUES: Set the denominator = 0, those \(x\) values are EXCLUDED from domain
What value of \(x\) makes \(\dfrac{x+4}{x^2 - x - 6}\) undefined?
Factor the denominator completely.
📖 해설
\(x^2 - x - 6 = (x-3)(x+2) = 0\)
따라서 \(x = 3\) 또는 \(x = -2\)일 때 분모가 0이 되어 정의 불능. 정답은 A.
따라서 \(x = 3\) 또는 \(x = -2\)일 때 분모가 0이 되어 정의 불능. 정답은 A.
Geometry
Angles · Triangles · Circles · Area · Proofs · Similarity
⚡ VERTICAL ANGLES: Opposite angles formed by two intersecting lines are EQUAL
Two lines intersect. One angle measures \(65°\). What is the measure of its vertical angle?
📖 해설
맞꼭지각(vertical angles)은 항상 서로 같습니다. \(65° = 65°\). 정답은 B.
⚡ EXTERIOR ANGLE THEOREM: exterior angle = sum of the two NON-adjacent interior angles
Quick Example
Interior angles \(40°\) and \(70°\): exterior angle at third vertex = \(40+70 = 110°\).
In triangle \(ABC\), \(\angle A = 48°\) and \(\angle B = 67°\). What is the exterior angle at \(C\)?
📖 해설
외각 정리: 외각 = 나머지 두 내각의 합
\(48° + 67° = 115°\). 정답은 C.
(참고: 내각 \(\angle C = 180 - 48 - 67 = 65°\), 외각 = \(180 - 65 = 115°\) 로도 확인 가능)
\(48° + 67° = 115°\). 정답은 C.
(참고: 내각 \(\angle C = 180 - 48 - 67 = 65°\), 외각 = \(180 - 65 = 115°\) 로도 확인 가능)
⚡ PYTHAGOREAN: \(a^2 + b^2 = c^2\) — \(c\) is ALWAYS the hypotenuse (longest side)
A right triangle has legs of length \(9\) and \(12\). What is the length of the hypotenuse?
📖 해설
\(c^2 = 9^2 + 12^2 = 81 + 144 = 225\)
\(c = \sqrt{225} = 15\). (3-4-5 삼각형의 3배!) 정답은 B.
\(c = \sqrt{225} = 15\). (3-4-5 삼각형의 3배!) 정답은 B.
⚡ ARC LENGTH: \(L = \dfrac{\theta}{360} \cdot 2\pi r\) — arc is just a fraction of the full circumference
A circle has radius \(10\). Find the arc length intercepted by a central angle of \(72°\). (Leave in terms of \(\pi\).)
📖 해설
\(L = \dfrac{72}{360} \times 2\pi(10) = \dfrac{1}{5} \times 20\pi = 4\pi\). 정답은 C.
⚡ SIMILAR RATIO: Corresponding sides are PROPORTIONAL. Set up cross-multiply proportion.
Quick Example
Triangles with sides ratio \(3:5\). If small side = 6, large side = \(\frac{5}{3} \times 6 = 10\).
Two similar triangles have corresponding sides in ratio \(3:7\). If the shorter side of the larger triangle is \(21\), what is the corresponding side of the smaller triangle?
📖 해설
\(\dfrac{3}{7} = \dfrac{x}{21}\) → \(x = \dfrac{3 \times 21}{7} = 9\). 정답은 B.
⚡ COMPOSITE AREA: Break into simple shapes. ADD areas together (or SUBTRACT holes).
A rectangle is \(10 \times 8\). A semicircle with diameter \(8\) is attached to one of the shorter sides. What is the total area? (Use \(\pi \approx 3.14\).)
📖 해설
직사각형 넓이: \(10 \times 8 = 80\)
반원 넓이 (반지름 4): \(\dfrac{1}{2}\pi r^2 = \dfrac{1}{2}(3.14)(16) = 25.12\)
합계: \(80 + 25.12 = 105.12\). 정답은 B.
반원 넓이 (반지름 4): \(\dfrac{1}{2}\pi r^2 = \dfrac{1}{2}(3.14)(16) = 25.12\)
합계: \(80 + 25.12 = 105.12\). 정답은 B.
⚡ CO-INTERIOR (Same-side interior): These angles ADD up to \(180°\) (supplementary) — they are NOT equal!
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle measures \(112°\). What is the other co-interior angle?
📖 해설
동측내각(co-interior angles)은 평행선에서 합이 180°.
\(180° - 112° = 68°\). 정답은 C.
(혼동 주의: 엇각은 같고, 동측내각은 보각 관계!)
\(180° - 112° = 68°\). 정답은 C.
(혼동 주의: 엇각은 같고, 동측내각은 보각 관계!)
⚡ CYLINDER VOLUME: \(V = \pi r^2 h\) — base area × height. Don't forget to SQUARE the radius!
Find the volume of a cylinder with radius \(5\) cm and height \(9\) cm. (Leave in terms of \(\pi\).)
📖 해설
\(V = \pi r^2 h = \pi (5)^2 (9) = 225\pi\) cm³. 정답은 B.
(자주 실수: \(5 \times 9 = 45\)로 계산해 A를 고르는 경우 → 반지름을 제곱해야!)
(자주 실수: \(5 \times 9 = 45\)로 계산해 A를 고르는 경우 → 반지름을 제곱해야!)
⚡ INSCRIBED ANGLE: inscribed angle = \(\dfrac{1}{2}\) × intercepted arc. Central angle = full arc.
Quick Example
Arc = \(80°\) → inscribed angle = \(40°\). Arc = \(180°\) (semicircle) → inscribed angle = \(90°\).
A central angle in a circle measures \(130°\). What is the inscribed angle that intercepts the same arc?
📖 해설
중심각 = 호의 크기 = \(130°\)
원주각 = \(\dfrac{1}{2} \times 130° = 65°\). 정답은 C.
원주각 = \(\dfrac{1}{2} \times 130° = 65°\). 정답은 C.
⚡ 30-60-90 SIDES: short leg \(: \text{long leg} : \text{hyp} = 1 : \sqrt{3} : 2\)
Quick Example
If hypotenuse = 10 in a 30-60-90: short leg = 5, long leg = \(5\sqrt{3}\).
In a 30-60-90 triangle, the side opposite the \(30°\) angle is \(7\). What is the hypotenuse?
📖 해설
30-60-90에서 비율: 짧은 변 : 빗변 = \(1:2\)
짧은 변(30° 맞은편)이 7이므로 빗변 = \(7 \times 2 = 14\). 정답은 C.
짧은 변(30° 맞은편)이 7이므로 빗변 = \(7 \times 2 = 14\). 정답은 C.