Complex numbers are numbers that have a real part and an imaginary part. The standard form is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Operations:

• Addition/Subtraction: Combine like terms (real parts with real parts, imaginary parts with imaginary parts).
• Multiplication: Use the distributive property and remember that \(i^2 = -1\).
• Division: Multiply the numerator and denominator by the conjugate of the denominator and simplify.

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Key Concepts:

• Degree: The highest exponent in the polynomial (e.g., the degree of \(2x^3 + 3x^2 - x + 5\) is 3).
• Factoring: Methods include factoring out the GCF, factoring trinomials, and using special products like the difference of squares.
• Polynomial Division: Two methods are long division and synthetic division.

A function is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output.

Key Concepts:

• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (y-values).
• Transformations: Includes translations (shifts), reflections, and stretches/compressions.

A quadratic equation is a second-degree polynomial equation of the form \(ax^2 + bx + c = 0\).

Solving Methods:

• Factoring: Express the quadratic as a product of binomials.
• Quadratic Formula: Use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.
• Completing the Square: Rewrite the quadratic in the form \((x + p)^2 = q\) and solve.

Discriminant: The expression \(b^2 - 4ac\) indicates the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root (a repeated root).
• If \(b^2 - 4ac < 0\), there are two complex roots.

Rational equations involve fractions with polynomials in the numerator and denominator.

Steps to Solve:

• Find a common denominator and multiply both sides by it to eliminate the fractions.
• Simplify and solve the resulting polynomial equation.
• Check for extraneous solutions by substituting back into the original equation.

Asymptotes: Identify vertical, horizontal, and slant asymptotes based on the degree of the numerator and denominator.

Exponential equations have the form \(a^x = b\), and logarithmic equations have the form \(\log_b(x) = y\).

Key Properties:

• \(a^m \cdot a^n = a^{m+n}\)
• \((a^m)^n = a^{mn}\)
• \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)

Solving Steps:

• For exponential equations, take the logarithm of both sides.
• For logarithmic equations, exponentiate both sides to remove the logarithm.
• Simplify and solve for the variable.

A linear function is a function of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Key Forms:

• Slope-Intercept Form: \(y = mx + b\)
• Point-Slope Form: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line.

A quadratic function is a function of the form \(y = ax^2 + bx + c\). It graphs as a parabola.

Key Forms:

• Standard Form: \(y = ax^2 + bx + c\)
• Vertex Form: \(y = a(x - h)^2 + k\), where (h, k) is the vertex of the parabola.
• Factored Form: \(y = a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots of the quadratic.

Polynomial functions are functions that are represented by polynomials. The degree of the polynomial determines the behavior of the graph at the ends (end behavior).

Key Concepts:

• End Behavior: The direction the graph moves as \(x\) approaches positive or negative infinity.
• Factoring and Roots: The roots of the polynomial correspond to the x-intercepts of the graph.
• The Fundamental Theorem of Algebra: Every polynomial equation of degree \(n\) has exactly \(n\) roots (counting multiplicities).

Rational functions are functions of the form \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) \neq 0\).

Asymptotes:

• Vertical Asymptotes: Occur where \(q(x) = 0\), and the function approaches infinity.
• Horizontal Asymptotes: Determined by the degrees of \(p(x)\) and \(q(x)\). If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• Slant Asymptotes: Occur when the degree of the numerator is one more than the degree of the denominator.

Exponential functions have the form \(f(x) = a \cdot b^x\), where \(a\) and \(b\) are constants, and \(b > 0\). Logarithmic functions are the inverse of exponential functions and have the form \(f(x) = \log_b(x)\).

Graphing:

• Exponential functions have a horizontal asymptote and increase or decrease rapidly.
• Logarithmic functions have a vertical asymptote and increase or decrease slowly.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The general term of an arithmetic sequence is given by \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Sum of n Terms:

• The sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2}(a_1 + a_n)\).

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general term of a geometric sequence is given by \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Sum of n Terms:

• The sum of the first \(n\) terms of a geometric sequence is \(S_n = a_1 \frac{1 - r^n}{1 - r}\) for \(r \neq 1\).

A series is the sum of the terms of a sequence. For arithmetic and geometric series, there are specific formulas to calculate the sum of the first \(n\) terms.

Infinite Geometric Series:

• An infinite geometric series converges to \(S = \frac{a_1}{1 - r}\) if \(|r| < 1\).

A parabola is the graph of a quadratic function. The standard form of a parabola is \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\).

Vertex Form:

• The vertex form of a parabola is \(y = a(x - h)^2 + k\), where (h, k) is the vertex.

A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where (h, k) is the center and \(r\) is the radius.

An ellipse is a set of all points in a plane where the sum of the distances from two fixed points (the foci) is constant. The standard form of an ellipse is \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where (h, k) is the center, \(a\) is the distance from the center to a vertex along the major axis, and \(b\) is the distance from the center to a vertex along the minor axis.

A hyperbola is a set of all points in a plane where the difference of the distances from two fixed points (the foci) is constant. The standard form of a hyperbola is \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\) (horizontal hyperbola) or \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\) (vertical hyperbola).

Asymptotes: The asymptotes of a hyperbola are the lines that the hyperbola approaches but never touches. For a hyperbola centered at (h, k), the asymptotes are given by the equations \(y - k = \pm \frac{b}{a}(x - h)\) (horizontal hyperbola) or \(y - k = \pm \frac{a}{b}(x - h)\) (vertical hyperbola).