SAT Math cheat sheet

Slope between two points: m = (y2 - y1)/(x2 - x1); parallel lines have equal slopes, and perpendicular lines have slopes whose product is -1.

Slope-intercept form y = mx + b has slope m and y-intercept b; point-slope form y - y1 = m(x - x1) passes through (x1, y1).

In standard form Ax + By = C, the slope is -A/B and the y-intercept is C/B.

A linear system in two variables has no solution when the lines have equal slopes but different intercepts, and infinitely many solutions when one equation is a nonzero multiple of the other.

Solve a linear system by substitution (replace one variable using the other equation) or by elimination (add or subtract multiples of the equations to cancel a variable).

A one-variable linear equation that simplifies to a true statement such as 0 = 0 has infinitely many solutions; one that simplifies to a false statement such as 0 = 5 has none.

For c >= 0, the equation |ax + b| = c splits into ax + b = c and ax + b = -c; if c < 0 there is no solution.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

In a linear model, the slope is the constant rate of change and the y-intercept is the starting value.

Quadratic formula: for ax^2 + bx + c = 0 with a != 0, x = (-b +/- sqrt(b^2 - 4ac)) / (2a).

The discriminant b^2 - 4ac gives two distinct real solutions when positive, one repeated real solution when zero, and no real solutions when negative.

For f(x) = ax^2 + bx + c, the axis of symmetry is x = -b/(2a) and the vertex lies on it; the vertex is a maximum when a < 0 and a minimum when a > 0.

Vertex form f(x) = a(x - h)^2 + k has vertex (h, k); in a(x - p)(x - q) form the x-intercepts are p and q and the axis of symmetry is halfway between them.

Sum of the roots of ax^2 + bx + c = 0 is -b/a and the product of the roots is c/a.

Exponential model y = a*b^x has initial value a; b > 1 gives growth and 0 < b < 1 gives decay, where b = 1 + r for rate r written as a decimal.

Compound interest: A = P(1 + r/n)^(nt), with principal P, annual rate r, n compounds per year, and t years.

Function composition: (f o g)(x) = f(g(x)); inverse functions satisfy f(f^-1(x)) = x and their graphs mirror across y = x.

Remainder theorem: dividing polynomial f(x) by (x - k) leaves remainder f(k); if f(k) = 0 then (x - k) is a factor of f(x).

For a rational function, vertical asymptotes occur where the denominator is zero and does not cancel with the numerator; compare numerator and denominator degrees to find any horizontal asymptote.

End behavior of a polynomial: even degree gives the same direction on both ends, odd degree gives opposite directions, and the leading coefficient controls upward or downward opening.

Percent change = (new value - old value) / old value x 100; a percent increase followed by the same percent decrease does not restore the original amount.

Set up proportions as a/b = c/d and solve by cross-multiplying: ad = bc.

Unit rate is found by dividing so the denominator becomes 1; use dimensional analysis and cancel matching units to convert measurements.

Mean = sum of values / number of values; median is the middle ordered value (average the two middle values when the count is even); mode is the most frequent value.

Range = maximum - minimum; standard deviation measures spread about the mean, so a larger standard deviation means more variability.

Adding a constant to every data value shifts the mean and median by that constant but leaves the standard deviation and range unchanged; multiplying every value by k multiplies the mean and standard deviation by k.

Probability = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) - P(A and B), and P(A and B) = P(A)P(B) when A and B are independent.

Conditional probability P(A | B) = P(A and B) / P(B), restricted to the outcomes inside event B.

Random sampling from a population supports generalizing results to that population; random assignment to treatment groups supports causal conclusions.

A larger random sample generally produces a smaller margin of error, and a reported estimate is the sample statistic plus or minus that margin.

Correlation between two variables does not by itself establish causation; a lurking variable can drive both.

Circle with center (h, k) and radius r: (x - h)^2 + (y - k)^2 = r^2; completing the square converts a general circle equation to this form.

Circle area = pir^2 and circumference = 2pir; arc length = (theta/360)(2pir) and sector area = (theta/360)(pir^2) for central angle theta in degrees.

In radians, 180 degrees = pi radians, arc length s = rtheta, and sector area = (1/2)r^2theta.

Pythagorean theorem: a^2 + b^2 = c^2 for a right triangle with hypotenuse c; common triples include 3-4-5, 5-12-13, and 8-15-17.

45-45-90 triangle sides are in ratio 1 : 1 : sqrt(2); 30-60-90 triangle sides are in ratio 1 : sqrt(3) : 2.

Right-triangle ratios: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent, and tan(theta) = sin(theta)/cos(theta).

Identity sin^2(theta) + cos^2(theta) = 1 holds for every angle, and sin(90 - theta) = cos(theta).

Similar triangles have equal corresponding angles and proportional corresponding sides; the ratio of their areas equals the square of the side ratio.

Volumes: cylinder pir^2h, cone (1/3)pir^2h, sphere (4/3)pir^3, pyramid (1/3)(base area)(height); sphere surface area is 4pi*r^2.