CLEP Precalculus key terms

A sum of terms in which each term is a constant multiplied by a variable raised to a whole-number power. Its degree equals the largest exponent with a nonzero coefficient.

The largest exponent appearing on the variable in a single-variable polynomial after like terms are combined. It caps the number of real zeros and of turning points.

Rewriting an expression as a product of simpler expressions whose multiplication restores the original. Common patterns include greatest common factor, difference of squares, and grouping.

A quotient of two polynomials, defined only where its denominator is nonzero. Simplifying requires canceling only factors shared by the entire numerator and denominator.

A number written a + bi, where a and b are real and i satisfies i squared equals negative one. Real numbers are the special case b = 0.

For a + bi, the number a - bi, found by reversing the sign of the imaginary part. Their product is the real number a squared plus b squared.

For a x squared + b x + c = 0 with a nonzero, the solutions are x = (-b ± sqrt(b squared - 4ac)) / (2a). It solves every quadratic equation, factorable or not.

The quantity b squared - 4ac from a quadratic equation. Its sign tells whether the two roots are real and distinct, real and equal, or a complex conjugate pair.

Rewriting a quadratic in the form a(x - h) squared + k by adding and subtracting a constant. It reveals the vertex and converts a circle equation to standard form.

If a product of factors equals zero, then at least one factor equals zero. It converts a factored equation into separate simpler equations to solve.

A value produced by solving a transformed equation that does not satisfy the original. It arises when both sides are squared or multiplied by an expression that can equal zero.

A statement such as |x - c| < r, meaning the distance from x to c is less than r. It becomes a compound inequality, here c - r < x < c + r.

A shorthand for sets of real numbers using brackets and parentheses, where a bracket includes an endpoint and a parenthesis excludes it. Infinity always takes a parenthesis.

A rule assigning each input in its domain exactly one output. Passing the vertical line test on a graph confirms the relation is a function.

The set of all inputs for which a function is defined. Common restrictions exclude values making a denominator zero or a radicand negative.

The set of all output values a function actually produces as its input runs through the domain. It is read from a graph along the vertical axis.

Applying one function to the output of another, written (f circle g)(x) = f(g(x)). The inner function's range must meet the outer function's domain.

The function undoing another, so that f(f inverse of x) equals x and vice versa. It exists only when the original function is one-to-one.

A function giving different outputs for different inputs, so no output repeats. Only such functions have inverses that are themselves functions.

A graph check: if every horizontal line meets the graph at most once, the function is one-to-one. Repeated heights signal a failed test.

A function satisfying f(-x) = f(x) for every x in its domain. Its graph is symmetric about the vertical axis.

A function satisfying f(-x) = -f(x) for every x in its domain. Its graph has rotational symmetry about the origin.

How a function's outputs behave as inputs grow without bound in the positive or negative direction. For polynomials it is governed by the leading term.

A point where a graph crosses the horizontal axis, so the output is zero. Finding intercepts means solving f(x) = 0.

An input value making the function equal zero. Real zeros correspond to x-intercepts, and a zero's multiplicity affects whether the graph crosses or touches there.

A vertical line a graph approaches where the function grows without bound, typically where a denominator is zero without canceling. The graph never crosses it.

A horizontal line a graph approaches far from the origin. For a rational function it depends on comparing the degrees of numerator and denominator.

The highest or lowest point of a parabola, located at x = -b/(2a) for y = a x squared + b x + c. It marks the maximum or minimum output.

The vertical line through a parabola's vertex that maps each half of the graph onto the other. Its equation is x = -b/(2a).

A curve formed by slicing a double cone with a plane: circle, ellipse, parabola, or hyperbola. Each has a standard equation revealing its center or vertex.

The set of points whose distances to two fixed foci have a constant sum. Its standard equation has both squared terms positive with unequal denominators.

The set of points whose distances to two foci have a constant absolute difference. Its standard equation has one positive and one negative squared term.

The distance between points (x1, y1) and (x2, y2) is the square root of (x2 - x1) squared plus (y2 - y1) squared. It follows from the Pythagorean theorem.

The point halfway between two endpoints has coordinates that are the averages of the corresponding endpoint coordinates. It also locates centers of circles and ellipses.

An angle measure defined by arc length divided by radius, so a full turn is 2 pi radians. Converting to degrees multiplies by 180/pi.

The circle of radius one centered at the origin. A point at angle theta on it has coordinates cosine theta and sine theta, which defines those functions for all angles.

The acute angle between the terminal side of an angle and the horizontal axis. Its sine and cosine magnitudes give those of the original angle, with signs set by quadrant.

The smallest positive input shift leaving a periodic function's values unchanged. Sine and cosine have period 2 pi; tangent has period pi.

Half the difference between the maximum and minimum values of a sinusoidal function. In y = A sin(Bx) + D it equals the absolute value of A.

The horizontal translation of a sinusoidal graph from its usual position. In y = A sin(B(x - C)) + D the shift is C units.

In any triangle, the ratio of a side to the sine of its opposite angle is the same for all three sides. It solves triangles given an angle and its opposite side.

A generalization of the Pythagorean theorem: c squared = a squared + b squared - 2ab cos(C). It handles triangles given two sides and the included angle, or all three sides.

The distance along a circle's edge spanned by an angle. For radius r and angle theta in radians, it equals r times theta.

A function of the form f(x) = a times b to the x, with positive base b not equal to one. It grows or decays by a constant factor per unit input.

The exponent to which a base must be raised to produce a given number. It is the inverse of the exponential with the same base.

The time required for a quantity undergoing exponential decay to fall to half its current amount. It is constant regardless of starting size.

Every polynomial of degree n at least one has at least one complex zero, so counting multiplicity it has exactly n complex zeros. Real roots may be fewer.