SAT Math flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Slope formula

For two points (x1, y1) and (x2, y2), the slope m equals (y2 - y1) divided by (x2 - x1), measuring vertical change per unit of horizontal change.

Slope-intercept form

The equation y = mx + b describes a line with slope m and y-intercept b, where b is the y-value at which the line crosses the y-axis.

Point-slope form

The equation y - y1 = m(x - x1) describes the line through the point (x1, y1) with slope m, and is used when a point and slope are known.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals of each other, so the product of the two slopes equals -1.

Parallel line slopes

Two distinct nonvertical lines are parallel exactly when they share the same slope, which means the lines never intersect anywhere in the coordinate plane.

System with no solution

A linear system has no solution when the two lines have equal slopes but different y-intercepts, making them parallel lines that never intersect.

Infinitely many solutions

A linear system has infinitely many solutions when one equation is a nonzero multiple of the other, so both equations describe exactly the same line.

Absolute value equation split

For c greater than or equal to zero, the equation |ax + b| = c is equivalent to the two equations ax + b = c and ax + b = -c.

Quadratic formula

For ax^2 + bx + c = 0 with a not equal to zero, the solutions are x = (-b plus or minus the square root of b^2 - 4ac) divided by 2a.

Discriminant

The expression b^2 - 4ac under the quadratic formula's radical: positive gives two real solutions, zero gives one repeated real solution, negative gives no real solutions.

Vertex of a parabola

For f(x) = ax^2 + bx + c, the vertex has x-coordinate -b/(2a); substituting that value into f gives the maximum or minimum output.

Vertex form

The form f(x) = a(x - h)^2 + k displays the vertex (h, k) and axis of symmetry x = h, with a > 0 opening upward and a < 0 opening downward.

Exponential growth model

The model y = a*b^x has starting value a; when b is greater than 1 the quantity grows by factor b each unit, and b equals 1 plus the decimal rate.

Compound interest formula

A = P(1 + r/n)^(nt) gives the balance after t years, where P is principal, r the annual rate as a decimal, and n the compounding periods per year.

Function composition

The composition (f o g)(x) equals f(g(x)), meaning apply g first and then apply f to that output; composition order generally changes the result.

Remainder theorem

Dividing a polynomial f(x) by (x - k) leaves a remainder equal to f(k); if f(k) = 0, then (x - k) is a factor of f(x).

Percent change formula

Percent change equals the new value minus the old value, divided by the old value, then multiplied by 100; a positive result is an increase.

Mean of a data set

The mean is the sum of all data values divided by the number of values; adding the same constant to every value increases the mean by that constant.

Median

The median is the middle value when data are ordered from least to greatest, or the average of the two middle values when the count is even.

Standard deviation meaning

Standard deviation measures how far data values typically sit from the mean; a larger standard deviation indicates that the data are more spread out.

Probability of an event

The probability of an event equals favorable outcomes divided by total equally likely outcomes, always producing a number from 0 through 1 inclusive.

Independent events rule

Events A and B are independent when the probability of both equals P(A) times P(B), so knowing one occurred does not change the other's probability.

Margin of error

A survey estimate is reported as the sample statistic plus or minus a margin of error; larger random samples generally produce smaller margins of error.

Random assignment versus random sampling

Random assignment of subjects to treatment groups supports causal conclusions, while random sampling from a population supports generalizing results from the sample to that population.

Circle equation

A circle with center (h, k) and radius r has equation (x - h)^2 + (y - k)^2 = r^2, with r equal to the square root of the right side.

Arc length formula

Arc length equals (theta/360) times 2*pi*r for a central angle theta in degrees; using theta in radians, arc length is simply r times theta.

Pythagorean theorem

In a right triangle, a^2 + b^2 = c^2 where c is the hypotenuse; frequently used triples are 3-4-5, 5-12-13, and 8-15-17.

45-45-90 triangle

An isosceles right triangle has two legs of length x and a hypotenuse of length x times the square root of 2, giving side ratio 1 : 1 : sqrt(2).

30-60-90 triangle

Sides are in ratio 1 : sqrt(3) : 2; the side opposite 30 degrees is x, opposite 60 degrees is x*sqrt(3), and the hypotenuse is 2x.

Sine, cosine, tangent ratios

For an acute angle in a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

Similar triangle side ratios

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

Sphere volume

The volume of a sphere is (4/3)*pi*r^3 and its surface area is 4*pi*r^2, where r is the radius of the sphere.