ACCUPLACER AAF key terms

An equation in which every variable appears to the first power only, so its graph is a straight line. Solving it means finding the value(s) that make the two sides equal.

The steepness of a line, computed as the change in the vertical coordinate divided by the corresponding change in the horizontal coordinate between two points on the line.

The point where a graph crosses the vertical axis, occurring where the horizontal coordinate equals zero. For a line written as y = mx + b, it is the value b.

The point where a graph crosses the horizontal axis, occurring where the vertical coordinate equals zero. It is found by setting y equal to zero and solving.

The way of writing a line as y = mx + b, where m is the slope and b is the vertical intercept. It lets you graph a line quickly from two numbers.

A way of writing a line as y − y₁ = m(x − x₁), using a known point (x₁, y₁) and the slope m. Useful when you know one point and the slope.

Writing a linear equation as Ax + By = C, where A, B, and C are constants and A and B are not both zero. Handy for finding both intercepts quickly.

Two lines in the same plane that never meet. They have equal slopes but different vertical intercepts, so they never share a solution point.

Two lines that meet at a right angle. Their slopes are negative reciprocals of each other, so the product of the slopes equals −1.

Two or more equations considered together, whose solution is the set of values satisfying all of them at once. Graphically it is where the graphs intersect.

A technique for solving a system by solving one equation for a variable and replacing that variable in the other equation. It reduces the system to a single equation.

A technique for solving a system by adding or subtracting multiples of the equations so one variable cancels. It leaves an equation in one variable.

A statement comparing two expressions with symbols such as <, >, ≤, or ≥. Its solution is a range of values rather than a single number.

A compact way to write a range of numbers using brackets and parentheses, such as [2, 5) meaning 2 included up to 5 excluded. Parentheses mean the endpoint is not included.

The collection of all values that make an equation or inequality true. For a line, it is every point on the line; for an inequality, it is a region.

An equation of the form ax² + bx + c = 0 with a not equal to zero. It can have two, one, or no real solutions depending on its discriminant.

Rewriting an expression as a product of simpler expressions whose multiplication gives the original. It is a key tool for solving polynomial equations set equal to zero.

The rule that if a product of factors equals zero, then at least one factor must equal zero. It converts a factored equation into simpler equations to solve.

The formula x = (−b ± √(b² − 4ac)) / (2a) giving the solutions of ax² + bx + c = 0. It works for every quadratic equation.

The expression b² − 4ac taken from a quadratic equation. Its sign tells whether there are two real solutions, one repeated real solution, or no real solutions.

A method of rewriting a quadratic expression as a perfect square plus a constant. It is used to solve quadratics and to convert equations of circles to standard form.

The highest or lowest point on a parabola. For y = ax² + bx + c, its horizontal coordinate is −b/(2a).

The U-shaped curve that is the graph of a quadratic function. It opens upward when the leading coefficient is positive and downward when negative.

The vertical line through the vertex of a parabola that divides it into mirror-image halves. Its equation is x = −b/(2a) for a quadratic in standard form.

A fraction whose numerator and denominator are polynomials. Its domain excludes values that make the denominator zero.

A value of the variable that makes a denominator equal zero, so the expression has no meaning there. These values are excluded from the domain.

The smallest expression divisible by each denominator in a set of fractions. Multiplying every term by it clears fractions from a rational equation.

A value produced during solving that does not actually satisfy the original equation. It often arises after squaring both sides or multiplying by an expression that can be zero.

An expression containing a root, such as a square root or cube root. It can often be simplified by removing perfect-power factors from the radicand.

An equation in which the variable appears inside a radical. It is usually solved by isolating the radical and raising both sides to a power, then checking for extraneous solutions.

Rewriting a fraction so no radical remains in the denominator, usually by multiplying numerator and denominator by a suitable radical or conjugate.

A pair of binomials that differ only in the sign between their terms, such as a + b and a − b. Their product contains no radical or imaginary term.

A number written as a + bi, where a and b are real and i is the square root of −1. It appears when a quadratic has no real solutions.

A sum of terms of the form a·xⁿ where n is a whole number and a is a constant. Its degree is the largest exponent with a nonzero coefficient.

The largest exponent on the variable in a polynomial, or the sum of exponents in a term of several variables. It largely determines the graph's shape and end behavior.

How the values of a function behave as the input grows very large positive or very large negative. For polynomials it depends on degree and leading coefficient.

An input value for which the function equals zero. On a graph, zeros are the horizontal intercepts; for a polynomial they correspond to factors.

The rule that a polynomial has a factor (x − r) exactly when r is a zero of the polynomial. It links factoring with intercepts and division.

The rule that dividing a polynomial by (x − r) leaves a remainder equal to the polynomial evaluated at r. It gives a quick way to test candidate zeros.

A function of the form f(x) = a·bˣ with b positive and not equal to 1. It grows or decays by a constant factor for each equal increase in input.

The exponent to which a base must be raised to produce a given number. Written log_b(x) = y means bʸ = x, so logarithms undo exponentials.

The rule log_b(x) = log_c(x) / log_c(b), which lets you evaluate a logarithm in any base using a calculator. It converts between bases.

A logarithm with base e, written ln(x), where e is about 2.718. It is the inverse of the natural exponential function eˣ.

The constant multiplier b in an exponential model a·bˣ. A factor above 1 signals growth; a factor between 0 and 1 signals decay.

A unit of angle measure based on arc length, where one full circle is 2π radians. Converting from degrees multiplies by π/180.

The circle of radius 1 centered at the origin. Points on it give the cosine and sine of an angle as their coordinates.

A trigonometric function that, for an angle in a right triangle, equals the opposite side divided by the hypotenuse. On the unit circle it is the vertical coordinate.

A trigonometric function that, for an angle in a right triangle, equals the adjacent side divided by the hypotenuse. On the unit circle it is the horizontal coordinate.

A trigonometric function equal to sine divided by cosine, or opposite over adjacent in a right triangle. It is undefined where cosine equals zero.

Half the vertical distance between the maximum and minimum values of a periodic graph. For y = a·sin(x), the amplitude is the absolute value of a.

The length of the smallest input interval after which a periodic function repeats its values. For sine and cosine it is 2π radians.

The set of all input values for which a function or expression is defined. It excludes values causing division by zero or even roots of negatives.