Key Concepts in College Algebra: Functions, Graphs, Matrices, and Series

Functions and Their Graphs

Definition of a Function

A function is a relation in which each input (from the domain) is assigned exactly one output (in the range). Functions are often written as f(x).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Graph of a Function: The set of all points (x, f(x)) in the coordinate plane.

Example: The function $f(x) = x^2$ has domain $(-\infty, \infty)$ and range $[0, \infty)$.

Graphing Functions

To graph a function, plot points for several values of x and connect them smoothly if the function is continuous.

• Intercepts: Points where the graph crosses the axes.

• Increasing/Decreasing: A function is increasing where its graph rises as x increases, and decreasing where it falls.

Example: The graph of $f(x) = \sqrt{x}$ starts at (0,0) and increases slowly to the right.

Matrices and Systems of Equations

Matrix Notation and Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent and solve systems of equations.

• Matrix Addition: Add corresponding elements.

• Matrix Multiplication: Multiply rows by columns and sum the products.

• Identity Matrix: A square matrix with 1's on the diagonal and 0's elsewhere.

Example: For matrices $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ and $B = \begin{bmatrix}5 & 6 \\ 7 & 8\end{bmatrix}$, their sum is $A+B = \begin{bmatrix}6 & 8 \\ 10 & 12\end{bmatrix}$.

Solving Systems Using Matrices

Systems of linear equations can be written in matrix form as $AX = B$, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

• Row Reduction: Use elementary row operations to solve for variables.

• Determinant: For a 2x2 matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$, the determinant is $ad-bc$.

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

Sequences and Series

Sequences

A sequence is an ordered list of numbers, often defined by a formula for the nth term.

• Arithmetic Sequence: Each term differs from the previous by a constant (common difference d).

• Geometric Sequence: Each term is multiplied by a constant (common ratio r) to get the next term.

Example: Arithmetic: $a_n = a_1 + (n-1)d$; Geometric: $a_n = a_1 r^{n-1}$

Series and Summation Notation

A series is the sum of the terms of a sequence. Summation notation uses the Greek letter sigma ($\sum$) to represent the sum.

• Finite Series: $\sum_{k=1}^n a_k$

• Infinite Series: $\sum_{k=1}^\infty a_k$

Example: $\sum_{k=1}^4 k = 1 + 2 + 3 + 4 = 10$

Square Roots and Radicals

Square Roots

The square root of a number x is a value that, when multiplied by itself, gives x. The principal square root is denoted $\sqrt{x}$.

• Properties: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, $\sqrt{a^2} = |a|$

Example: $\sqrt{16} = 4$

Graphs and Data Representation

Scatter Plots and Graph Interpretation

A scatter plot is a graph of plotted points that show the relationship between two sets of data. It is useful for identifying trends, patterns, or correlations.

• Positive Correlation: As one variable increases, so does the other.

• Negative Correlation: As one variable increases, the other decreases.

• No Correlation: No apparent relationship between variables.

Example: Plotting students' study hours vs. test scores to see if more study leads to higher scores.

Summary Table: Key Algebraic Concepts

Concept	Definition	Example
Function	Relation assigning each input one output	$f(x) = x^2$
Matrix	Rectangular array of numbers	$\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$
Sequence	Ordered list of numbers	2, 4, 6, 8, ...
Series	Sum of sequence terms	$2 + 4 + 6 + 8$
Square Root	Value which, when squared, gives the original number	$\sqrt{9} = 3$