AoPS Intermediate Algebra Chapter 1-5

Systems of Equations

• A consistent system has at least one solution.
• An inconsistent system has no solutions.
• A dependent system has equations that are linear combinations of each other.
• An independent system has equations that are not linear combinations of one another.

Solutions and Functions

• An extraneous solution is a valid solution for a squared equation but doesn't satisfy the original equation.
• The domain of a function includes all possible input values that produce an output.
• The range of a function encompasses all possible output values.

Graphing Functions

• The vertical line test determines if a curve represents a function by checking if vertical lines intersect it at most once.
• Functions f and g are mutual inverses if ( g(f(x)) = x ) and ( f(g(x)) = x ) for all ( x ) in their domain.

Graph Transformations

• The graph of ( y = f(x) + k ) shifts vertically by ( k ) units.
• The graph of ( y = f(x + k) ) shifts horizontally to the left by ( k ) units.
• The graph of ( y = -f(x) ) reflects over the x-axis.
• The graph of ( y = f(-x) ) reflects over the y-axis.
• The graph of ( y = kf(x) ) scales vertically by a factor of ( k ).
• The graph of ( y = f(kx) ) scales horizontally by a factor of ( 1/k ).

Functions and Magnitudes

• The composition of two functions involves substituting the output of the first function into the second (e.g., ( f(g(x)) )).
• The conjugate of ( a + bi ) is ( a - bi ).
• The magnitude of ( a + bi ) is ( \sqrt{a^2 + b^2} ).

Quadratic Equations

• The discriminant of a quadratic ( ax^2 + bx + c ) is given by ( b^2 - 4ac ).
• If the discriminant equals 0, there is one root.
• If the discriminant is positive, the roots are real.
• If the discriminant is negative, the roots form a non-real conjugate pair.
• Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots: ( r+s = -b/a ) and ( rs = c/a ).

Geometric Structures

• A locus is a set of points satisfying specific conditions.
• A parabola is defined as points equidistant from a directrix and a focus.
• The axis of symmetry in a parabola is the line through the focus that is perpendicular to the directrix.
• A circle is the set of points at a fixed distance (radius) from a center.
• The standard form of a circle is expressed as ( (x-h)^2 + (y-k)^2 = r^2 ).
• An ellipse is defined where the sum of distances from two foci remains constant.
• The standard form of an ellipse is ( (x-h)^2/a^2 + (y-k)^2/b^2 = 1 ).
• The major axis connects the vertices of an ellipse.
• The minor axis is shorter and perpendicular to the major axis.
• The distance between foci can be derived from the major and minor axis lengths using the equation: ( (major\ axis\ length)^2 = (minor\ axis\ length)^2 + (distance\ between\ the\ foci)^2 ).

Hyperbola Properties

• The asymptotes of a hyperbola are lines approached by the hyperbola but never intersected, described by the equation ( (y-k)/a = \pm (x-h)/b ).