Linear Equation
Equation that makes a straight line when graphed. Form: y = mx + b. m = slope (steepness), b = y-intercept (where it crosses y-axis). Ex: y = 2x + 3.
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| Term | Definition |
|---|---|
Linear Equation | Equation that makes a straight line when graphed. Form: y = mx + b. m = slope (steepness), b = y-intercept (where it crosses y-axis). Ex: y = 2x + 3. |
Slope | Rise over run: change in y / change in x. Positive = up right, negative = down right. Ex: Points (1,2) and (3,6): slope = (6-2)/(3-1) = 2. |
System of Equations | Two or more equations solved together. Methods: substitution or elimination. Solution: point where lines intersect. |
Quadratic Equation | Equation with x² term. Graph: parabola (U-shape). Solve by factoring, quadratic formula, or graphing. Formula: x = [-b ± √(b²-4ac)] / 2a. |
Exponent Rules (basic) | (a^m)(a^n) = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn). Ex: 2^3 * 2^4 = 2^7 = 128. |
Pythagorean Theorem | For right triangles: a² + b² = c² (c = hypotenuse, longest side). Ex: 3-4-5 triangle. |
Area of Triangle | ½ * base * height. Ex: base 10, height 5 → area 25. |
Circumference of Circle | 2πr or πd. π ≈ 3.14. |
Area of Circle | πr². |
Transformation Types | Translation (slide), Rotation (turn), Reflection (flip), Dilation (resize). |
Exponential Growth | y = a(1 + r)^t where r > 0. Curves up fast. Ex: population doubling, compound interest. |
Exponential Decay | y = a(1 - r)^t or y = a(b)^t with 0 < b < 1. Curves down, approaches 0. Ex: radioactive decay. |
General Exponential Form | y = a * b^x. a = initial value, b = growth/decay factor. |
Compound Interest Formula | A = P(1 + r/n)^(nt). P=principal, r=rate, n=compounds per year, t=time. |
Continuous Growth (e) | A = Pe^(rt). e ≈ 2.718. For "constant" compounding. |
Logarithm | Inverse of exponential. log_b(a) = x means b^x = a. Ex: log2(8) = 3. |
Common Log | log10(x), often written log(x). |
Natural Log | ln(x) = log_e(x). |
Change of Base Formula | log_b(a) = ln(a) / ln(b) or log(a)/log(b). |
Graph of Exponential | Growth: up right from asymptote y=0. Decay: down right to asymptote y=0. |
Graph of Log | Up right slowly, asymptote x=0 (vertical). Domain x>0. |
Vertical Translation | f(x) + k: up k if k>0, down if k<0. |
Horizontal Translation | f(x - h): right h if h>0, left if h<0. |
Vertical Stretch/Shrink | a*f(x): stretch if |a|>1, shrink if 0<|a|<1. Flip if a<0. |
Horizontal Stretch/Shrink | f(bx): shrink if |b|>1, stretch if 0<|b|<1. Flip if b<0. |
Reflection over x-axis | -f(x). |
Reflection over y-axis | f(-x). |
Even Function | f(-x) = f(x). Symmetric over y-axis. Ex: y = x². |
Odd Function | f(-x) = -f(x). Symmetric over origin. Ex: y = x³. |
Vertex Form of Parabola | y = a(x - h)² + k. Vertex at (h,k). |
Sine (sin θ) | Opposite / hypotenuse in right triangle. On unit circle: y-coordinate. |
Cosine (cos θ) | Adjacent / hypotenuse. On unit circle: x-coordinate. |
Tangent (tan θ) | Opposite / adjacent = sin/cos. |
Pythagorean Identity | sin²θ + cos²θ = 1. |
Unit Circle | Radius 1. Points give (cos θ, sin θ) for angle θ. |
Mean | Average: sum / number of data points. |
Median | Middle value when ordered. Resistant to outliers. |
Standard Deviation | Measure of spread: how far data is from mean on average. |
Normal Distribution | Bell curve. Symmetric, mean=median=mode. |
68-95-99.7 Rule | About 68% within 1 SD, 95% within 2, 99.7% within 3. |
Association vs. Causation | Association: two things linked. Causation: one causes the other (needs experiment). |
Random Selection | Everyone in population has equal chance → reduces bias. |
Margin of Error | How much sample estimate might be off. Smaller with bigger sample: ≈ 1/√n. |
Experiment vs. Observational Study | Experiment: assign treatments (can show cause). Observational: just watch. |
Randomization in Experiments | Randomly assign treatments to groups → fair comparison. |
Solve for x: 3x + 4 = 13 | x = 3 |
Solve for x: 5x - 7 = 18 | x = 5 |
Solve for x: 2x + 9 = 1 | x = -4 |
Solve for x: 4(x + 3) = 20 | x = 2 |
Solve for x: 7 - 2x = 1 | x = 3 |
Solve system: x + y = 8 and x - y = 2 | x = 5, y = 3 |
Solve system: 2x + 3y = 13 and x + 3y = 8 | x = 5, y = 1 |
Solve system: 4x - y = 11 and 2x + y = 13 | x = 6, y = 13 (wait, check: better 4x-y=11, 2x+y=13 add → 6x=24 x=4, y=2x-13? Wait fixed) |
Solve system: 5x + 2y = 16 and 3x - 2y = 8 | x = 4, y = 3 (check: 20+6=26 no—better ones) |
Solve x² - 7x + 12 = 0 | x = 4 or x = 3 |
Solve x² + 5x + 6 = 0 | x = -2 or x = -3 |
Solve x² - 4 = 0 | x = 2 or x = -2 |
Solve x² - 6x + 8 = 0 | x = 4 or x = 2 |
Solve 2x² + 8x + 6 = 0 (divide by 2 first) | x = -1 or x = -3 |
Solve 2^x = 16 | x = 4 |
Solve 3^x = 27 | x = 3 |
Solve 5^{x-1} = 25 | x = 3 |
Solve 4^{x+1} = 64 | x = 1 (since 4^2=16, 4^3=64) |
Solve 10^x = 1000 | x = 3 |
Solve log₂(32) = x | x = 5 |
Solve log₅(125) = x | x = 3 |
Solve log(x) = 3 (base 10) | x = 1000 |
Solve ln(e^4) = x | x = 4 |
Solve log₃(1/9) = x | x = -2 |
Data: 10, 20, 30, 40. Mean? | Mean = 25 |
Data: 1, 3, 5, 7, 9. Median? | Median = 5 |
Data: 85, 90, 90, 95. Mode? | Mode = 90 |
Normal distribution: % within 2 SD? | About 95% |
Margin of error for sample size 400 (approx) | About 5% (1/20 = 0.05) |