Key Topics in College Algebra: Parabolas, Matrices, Functions, Trigonometry, and Sequences

Equation of a Parabola

Standard Equation and Graphical Features

The parabola is a fundamental conic section in algebra, defined by a quadratic equation. Understanding its standard equation and graphical features is essential for analyzing its properties and applications.

• Standard Equation: The standard form of a parabola opening upwards or downwards is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

• Focus and Directrix: Every parabola has a focus (a fixed point) and a directrix (a fixed line). The set of all points equidistant from the focus and directrix forms the parabola.

• Vertex: The vertex is the turning point of the parabola, located at $(h, k)$ in the standard equation.

• Axis of Symmetry: The line $x = h$ is the axis of symmetry for the parabola.

• Example: For $y = 2(x - 1)^2 + 3$, the vertex is at $(1, 3)$, the axis of symmetry is $x = 1$, and the parabola opens upwards.

Matrices

Basic Operations and Applications

Matrices are rectangular arrays of numbers that are used to represent and solve systems of equations, among other applications in algebra.

• Addition and Subtraction: Matrices of the same size can be added or subtracted by adding or subtracting their corresponding elements.

• Multiplying with a Scalar: Each entry of a matrix is multiplied by the scalar value.

• Multiplication of Matrices: The product of two matrices $A$ and $B$ is defined if the number of columns in $A$ equals the number of rows in $B$.

• Transpose of a Matrix: The transpose of matrix $A$, denoted $A^T$, is formed by swapping its rows and columns.

• Inverse of a Matrix: For a square matrix $A$, the inverse $A^{-1}$ satisfies $AA^{-1} = I$, where $I$ is the identity matrix.

Solving Systems of Equations with Matrices

• Matrix Inverse Method: For a system $AX = B$, the solution is $X = A^{-1}B$ if $A$ is invertible.

• Cramer's Rule: For a system of $n$ equations in $n$ unknowns, each variable is found using determinants: $x_i = \frac{\det(A_i)}{\det(A)}$.

• Gauss-Jordan Elimination: This method transforms the augmented matrix to reduced row-echelon form to find the solution.

• Example: Solve $\begin{cases} x + y = 2 \\ 2x - y = 1 \end{cases}$ using matrices.

Functions

Domains, Combinations, and Inverses

Functions describe relationships between sets of numbers. Understanding their domains, combinations, and inverses is crucial for algebraic analysis.

• Domain: The set of all possible input values ($x$) for which the function is defined.

• Combination of Functions: Functions can be added, subtracted, multiplied, or divided to form new functions.

• Composition of Functions: The composition $f(g(x))$ applies $g$ first, then $f$ to the result.

• Inverse of Functions: The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$, such that $f(f^{-1}(x)) = x$.

• Range: The set of all possible output values ($y$) of the function.

• Example: If $f(x) = 2x + 3$ and $g(x) = x^2$, then $f(g(x)) = 2x^2 + 3$.

Trigonometry

Basic Rules and Applications

Trigonometry deals with the relationships between the angles and sides of triangles, especially right-angled triangles.

• Pythagoras Rule: In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

• Finding Missing Angles: Use trigonometric ratios (sine, cosine, tangent) to find unknown angles in right triangles.

• Six Basic Trig Ratios: Sine, cosine, tangent, cosecant, secant, and cotangent.

• Solving Simple Trig Equations: Equations involving trigonometric functions can be solved for unknown angles or sides.

• Verifying Trig Identities: Show that two expressions are equivalent using algebraic and trigonometric properties.

• Inverse Trig Functions: Used to find angles when given a trigonometric ratio, e.g., $\sin^{-1}(x)$.

• Example: If $\sin \theta = 0.5$, then $\theta = 30^\circ$ or $\theta = 150^\circ$ (in degrees).

Sequences and Series

Arithmetic and Geometric Progressions

Sequences are ordered lists of numbers, and series are the sums of sequences. Two important types are arithmetic and geometric progressions.

• Arithmetic Progression (AP): Each term differs from the previous by a constant difference $d$. General term: $a_n = a_1 + (n-1)d$.

• Geometric Progression (GP): Each term is multiplied by a constant ratio $r$. General term: $a_n = a_1 r^{n-1}$.

• Example (AP): $2, 5, 8, 11, ...$ with $d = 3$.

• Example (GP): $3, 6, 12, 24, ...$ with $r = 2$.