Self-Study Worksheet

Master the Essentials

Key problems with memory tricks. Easy enough to start, smart enough to remember.

0 / 20 answered
Algebra 2
10 Essential Problems
1
Quadratic Functions Warm-up
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VERTEX FORM: $y = a(x-h)^2 + k$ → vertex is $(h, k)$. Watch the sign: $y=(x-\mathbf{3})^2$ means $h=+3$, NOT $-3$!
The parabola $y = 2(x - 3)^2 - 5$ has vertex at point $(h,k)$.
What is the value of $h + k$?
Quick Example
$y = (x+2)^2 - 7$ → vertex $= (-2,\,-7)$ → sum $= -9$
Answer:
2
Quadratic Formula Warm-up
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DISCRIMINANT: $b^2 - 4ac$  →  positive = 2 roots, zero = 1 root, negative = no real roots. Memorize: "POS-ONE-NEG"
For $3x^2 - 6x + 3 = 0$, how many distinct real solutions exist?
Hint: compute the discriminant $b^2 - 4ac$ first.
Quick Example
$x^2 - 4x + 4 = 0$: $\Delta = 16 - 16 = 0$ → exactly 1 solution ($x=2$)
Answer (number):
3
Polynomial Division Think
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REMAINDER THEOREM: remainder of $f(x) \div (x-c)$ = $f(c)$. Just plug in! No long division needed.
$f(x) = x^3 - 2x^2 + x - 4$ is divided by $(x - 2)$.
What is the remainder?
Quick Example
$f(x) = x^2 - 3x + 1$ divided by $(x-1)$: just compute $f(1) = 1 - 3 + 1 = -1$. Remainder = $-1$.
Remainder =
4
Rational Expressions Tricky!
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HOLE vs ASYMPTOTE: if the same factor cancels → HOLE. If it stays in denominator → VERTICAL ASYMPTOTE. Cancel first, always!
Simplify: $\dfrac{x^2 - 9}{x - 3}$
What is the simplified expression? (Type in the form "x + n")
Quick Example
$\dfrac{x^2-4}{x-2} = \dfrac{(x-2)(x+2)}{x-2} = x+2$  (with a hole at $x=2$)
Simplified:
5
Exponential Functions Warm-up
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GROWTH vs DECAY: $y = ab^x$ → $b > 1$ = growth, $0 < b < 1$ = decay. The y-intercept is always $a$ (plug in $x=0$).
A bacteria population starts at 500 and triples every hour.
Which equation models this? After 3 hours, what is the population?
Quick Example
Start = 200, doubles every hour → $y = 200 \cdot 2^x$. After 4 hrs: $200 \cdot 2^4 = 3200$.
Population at $t=3$:
6
Logarithms Think
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LOG = EXPONENT: $\log_b x = y \Leftrightarrow b^y = x$. Read it: "log base $b$ of $x$ equals the power". Swap log ↔ exponent to solve!
Solve for $x$:   $\log_2(x) = 5$
Quick Example
$\log_3(x) = 4$ → $3^4 = x$ → $x = 81$
$x$ =
7
Sequences & Series Tricky!
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ARITHMETIC vs GEOMETRIC: Arithmetic = add/subtract same number (common difference $d$). Geometric = multiply same number (common ratio $r$). Check: subtract consecutive terms vs divide.
A geometric sequence has first term $a_1 = 4$ and common ratio $r = 3$.
What is the 5th term $a_5$?
Quick Example
$a_1 = 2,\; r = 5$: $\quad a_n = 2 \cdot 5^{n-1}$ → $a_4 = 2 \cdot 5^3 = 250$
$a_5$ =
8
Radical & Rational Exponents Tricky!
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RATIONAL EXPONENT: $x^{m/n} = (\sqrt[n]{x})^m = \sqrt[n]{x^m}$. Think: denominator = root index, numerator = power. "Down-Root, Up-Power"
Simplify: $\; 27^{2/3}$
Step 1: find $\sqrt[3]{27}$. Step 2: raise to the 2nd power.
Quick Example
$8^{2/3}$: $\sqrt[3]{8} = 2$, then $2^2 = 4$. ✓
Answer =
9
Systems of Equations Think
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SUBSTITUTION SHORTCUT: isolate the simplest variable first → substitute → solve. If both equations are linear, pick the one with no fraction or coefficient.
Solve the system:
$\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}$
What is $x + y$?
Quick Example
$y = x+2$ and $2x+y=8$: sub → $2x+(x+2)=8$ → $x=2, y=4$ → sum $= 6$
$x + y$ =
10
Complex Numbers Tricky!
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POWERS OF $i$: $i^1=i,\; i^2=-1,\; i^3=-i,\; i^4=1$, then repeats! Divide the exponent by 4, use the remainder only. "i-cycle-4"
Multiply: $(3 + 2i)(1 - i)$
Express the answer in the form $a + bi$. What is the real part $a$?
Quick Example
$(2+i)(1-i) = 2 - 2i + i - i^2 = 2 - i - (-1) = 3 - i$   → real part = 3
Real part $a$ =
Geometry 2
10 Essential Problems
1
Circle — Arc Length Warm-up
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ARC = FRACTION of circle: Arc length $= \dfrac{\theta}{360} \times 2\pi r$. Just multiply the full circumference by the angle fraction. "Slice the pie"
90° r
A circle has radius $r = 10$. Find the arc length of a 90° central angle.
Express your answer in terms of $\pi$ (e.g., write "5pi" for $5\pi$).
Quick Example
$r=6,\; \theta=60°$: arc $= \dfrac{60}{360} \times 2\pi(6) = \dfrac{1}{6} \times 12\pi = 2\pi$
Arc length =
2
Pythagorean Theorem Warm-up
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HYPOTENUSE is always the LONGEST side: $a^2 + b^2 = c^2$ where $c$ = hypotenuse (opposite the right angle). If two legs given → find $c$. If one leg + hypotenuse → subtract!
In a right triangle, one leg is $8$ and the hypotenuse is $17$.
What is the length of the other leg?
Quick Example
Leg = 6, hyp = 10: $6^2 + b^2 = 10^2$ → $b^2 = 64$ → $b = 8$. (Classic 6-8-10 triple!)
Other leg =
3
Similar Triangles Think
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SCALE FACTOR: corresponding sides are proportional. Set up the ratio: $\dfrac{\text{small}}{\text{large}} = \dfrac{\text{small}}{\text{large}}$. Cross-multiply to solve. "Match and cross!"
Two similar triangles: the smaller has sides $3, 4, 5$ and the larger has its shortest side equal to $9$.
What is the longest side of the larger triangle?
Quick Example
Small sides: $2, 3, 4$. Largest small side = 4. Scale = 2 → large side = $4 \times 2 = 8$.
Longest side =
4
Circle — Inscribed Angle Tricky!
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INSCRIBED ANGLE = HALF the arc: inscribed angle $= \dfrac{1}{2} \times$ intercepted arc. Central angle = full arc. Don't mix them up! "Inscribed = Half"
An inscribed angle intercepts an arc of 140°.
What is the measure of the inscribed angle in degrees?
Quick Example
Intercepted arc = 80° → inscribed angle = $80° \div 2 = 40°$
Angle =
5
Volume — Cone & Cylinder Think
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CONE = ⅓ CYLINDER: $V_{\text{cone}} = \dfrac{1}{3}\pi r^2 h$, $\; V_{\text{cylinder}} = \pi r^2 h$. Same $r$ and $h$ → cone holds exactly $\frac{1}{3}$ as much. "Cone-Third"
A cone has radius $r = 3$ and height $h = 4$.
Find the volume. Express in terms of $\pi$ (e.g., "12pi").
Quick Example
$r=2, h=6$: $V = \dfrac{1}{3}\pi(4)(6) = 8\pi$
Volume =
6
Coordinate Geometry Warm-up
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MIDPOINT: $M = \left(\dfrac{x_1+x_2}{2},\; \dfrac{y_1+y_2}{2}\right)$ → just average the $x$'s and average the $y$'s. "Average both coordinates"
Points $A = (2, 6)$ and $B = (8, -2)$.
Find the midpoint $M$. What is the $y$-coordinate of $M$?
Quick Example
$A=(0,4), B=(6,8)$: $M = (3, 6)$
$y$-coordinate =
7
Parallel Lines & Transversals Tricky!
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ANGLE PAIRS: Alternate interior = equal (Z-shape). Co-interior (same-side) = supplementary (adds to 180°). Corresponding = equal (F-shape). Z=equal, C=180
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is $65°$.
What is the measure of the other co-interior angle?
Quick Example
Co-interior = 110° → other = $180°-110°=70°$
Angle =
8
Trigonometry — SOH CAH TOA Think
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SOH CAH TOA: $\sin = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$, $\; \cos = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$, $\; \tan = \dfrac{\text{Opposite}}{\text{Adjacent}}$. Label the triangle FIRST!
In a right triangle with hypotenuse $= 13$ and one angle $= 30°$.
What is the length of the side opposite the 30° angle?
Use $\sin 30° = 0.5$
Quick Example
Hyp = 10, angle = 30°: opposite = $10 \times \sin 30° = 10 \times 0.5 = 5$
Opposite side =
9
Circle — Equation Tricky!
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CIRCLE EQUATION: $(x-h)^2 + (y-k)^2 = r^2$. Center = $(h, k)$, radius = $r$. Watch signs: $(x+3)^2$ means center $x = -3$, not $+3$! "Opposite sign for center"
The equation of a circle is $(x-4)^2 + (y+1)^2 = 25$.
What is the radius?
Quick Example
$(x-1)^2 + (y-2)^2 = 49$ → center $(1,2)$, radius $= \sqrt{49} = 7$
$r$ =
10
Surface Area — Sphere Think
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SPHERE FORMULAS: Surface area $= 4\pi r^2$. Volume $= \dfrac{4}{3}\pi r^3$. Memorize: Surface = 4 circles. Volume = 4/3 pie r cubed. "Four-Four-Three"
A sphere has radius $r = 6$.
What is its surface area? Express in terms of $\pi$ (e.g., "144pi").
Quick Example
$r = 3$: SA $= 4\pi(3)^2 = 4\pi \times 9 = 36\pi$
Surface area =