TASC Mathematics Methods – Foundation Level 3 flashcards

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

Slope formula

The slope through two points (x1,y1) and (x2,y2) is (y2 - y1)/(x2 - x1). It measures vertical change per unit of horizontal change.

Quadratic formula

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

Discriminant

The value b^2 - 4ac from a quadratic. Positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Function definition

A rule assigning exactly one output to each input in its domain. A graph passes the vertical line test only when it is a function.

Vertex of a parabola

For y = ax^2 + bx + c, the vertex and axis of symmetry sit at x = -b divided by 2a. The vertex is the maximum or minimum.

Average rate of change

On the interval from a to b, it equals f(b) minus f(a), all divided by b minus a. It is the slope of the secant line.

Derivative as a limit

The derivative is the limit as h approaches zero of [f(x+h) - f(x)] divided by h. It gives the instantaneous rate of change at x.

Power rule

The derivative of x raised to n is n times x raised to n minus one, for any real exponent n. The derivative of a constant is zero.

Sign of the derivative

Where f'(x) is positive the function is increasing; where f'(x) is negative it is decreasing. Zero derivative marks a possible turning point.

Second derivative meaning

The second derivative describes how the slope changes. Positive means concave up, negative means concave down, and zero is a possible inflection point.

Velocity and acceleration

If s(t) gives position, velocity is the first derivative s'(t) and acceleration is the second derivative s''(t). Units follow from position per time.

Complement rule

The probability an event does not occur equals one minus the probability it does occur. Together the two probabilities sum to one.

Addition rule

For two events, the probability of A or B equals P(A) plus P(B) minus P(A and B). For mutually exclusive events the overlap term is zero.

Conditional probability

The probability of B given A equals the probability of both divided by the probability of A, provided P(A) is greater than zero.

Mean, median, mode

Mean is the sum divided by the count, median is the middle of ordered data, and mode is the most frequent value.

Normal distribution rule

In a normal distribution roughly 68 percent of values fall within one standard deviation of the mean and about 95 percent within two.

Linear versus exponential model

A linear model adds a constant amount each period. An exponential model multiplies by a constant factor, giving repeated percent change.

Compound interest formula

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

Exponential growth multiplier

For a percent change r written as a decimal, the per-period multiplier is 1 + r. Growth uses a multiplier above 1; decay uses one between 0 and 1.

Checking a model

Substitute known data into the model and compare predictions with observed values. A model is only reliable within the range of its data.

System with no solution

Two linear equations have no solution when their graphs are parallel: equal slopes but different y-intercepts. Equal slopes and intercepts mean infinitely many solutions.

Inverse function

The inverse reverses the original rule: if f(a) equals b, then the inverse of f at b equals a. Only one-to-one functions have inverses.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. Adding or subtracting never reverses it.

Chain rule

The derivative of f(g(x)) equals f'(g(x)) times g'(x). Differentiate the outer function, then multiply by the inner derivative.

Tangent line equation

At x = a the tangent slope is f'(a), and the line is y minus f(a) equals f'(a) times (x minus a).

Expected value

For a discrete random variable, expected value is the sum of each outcome multiplied by its probability. It is a long-run average, not a guaranteed result.

Combinations formula

The number of ways to choose r items from n without regard to order is n factorial divided by r factorial times (n minus r) factorial.

Interquartile range

IQR equals the third quartile minus the first quartile. It measures the spread of the middle half of the data and resists extreme values.

Correlation and causation

Correlation measures the strength and direction of a linear association. A strong correlation alone does not establish that one variable causes the other.

Break-even point

The input value where two models give equal output, such as cost equaling revenue. Find it by setting the two expressions equal and solving.

Direct variation

A relationship of the form y = kx, where k is a constant found from a known pair. Doubling x doubles y; the graph is a line through the origin.

Half-life formula

Remaining amount equals the initial amount times one half raised to the power t divided by the half-life. Time units must match.