College Algebra Final Review: Comprehensive Study Notes

Review of Algebra Fundamentals

Zeros of Polynomial Functions

Determining whether a given value is a zero of a polynomial is a foundational skill in algebra. A zero of a function is a value for which the function evaluates to zero.

• Substitution Method: Substitute the given value into the function and check if the result is zero.

• Example: For $f(x) = x^2 + 5x + 7$, substitute $x = 3$:

$f(3) = (3)^2 + 5(3) + 7 = 9 + 15 + 7 = 31$ Since $f(3) \neq 0$, 3 is not a zero of $f(x)$.

Equations and Graphs

Graphing Polynomial Functions

Graphing polynomials involves plotting points and identifying key features such as intercepts and end behavior.

• Choose the correct graph: Identify the degree and leading coefficient to determine the shape.

• Example: Graph $y = 3^x$ (see Exponential Functions below).

Long Division of Polynomials

Polynomial long division is used to divide one polynomial by another, often to determine if a value is a factor or root.

• Example: For $f(x) = x^2 - 3x - 4$, use long division to check if $x+1$ is a factor.

• If the remainder is zero, $x+1$ is a factor; otherwise, it is not.

Functions

Vertical and Horizontal Line Tests

These tests help determine whether a graph represents a function and whether it is one-to-one.

• Vertical Line Test: A graph is a function if any vertical line intersects the graph at most once.

• Horizontal Line Test: A function is one-to-one if any horizontal line intersects the graph at most once.

• One-to-One Functions: A function passes both the vertical and horizontal line tests.

Intercepts and Asymptotes

Finding intercepts and asymptotes is essential for graphing rational functions.

• Vertical Asymptote: Set the denominator equal to zero and solve.

• Domain: All real numbers except where the denominator is zero.

• x-intercepts: Set the numerator equal to zero and solve.

• y-intercept: Substitute $x = 0$ into the function.

• Example: For $f(x) = \frac{x^2 - 5x + 4}{x + 2}$:

Vertical Asymptote: $x + 2 = 0 \Rightarrow x = -2$ Domain: $(-\infty, -2) \cup (-2, \infty)$ x-intercepts: $x^2 - 5x + 4 = 0 \Rightarrow (x-1)(x-4) = 0 \Rightarrow x = 1, x = 4$ y-intercept: $f(0) = \frac{0^2 - 5(0) + 4}{0 + 2} = \frac{4}{2} = 2$

Polynomial and Rational Functions

Graphing Examples

• Exponential Function: $y = 3^x$

• Rational Function: $y = \frac{1}{x}$

• Logarithmic Function: $y = \log_{3}x$

Exponential and Logarithmic Functions

Exponential and Logarithmic Forms

Exponential and logarithmic expressions are closely related and can be converted between forms.

Exponential Form	Logarithmic Form
$3^2 = 9$	$\log_3 9 = 2$
$e^1 = e$	$\ln e = 1$

Additional info: Always check that the base of a logarithm is positive and not equal to 1.

Change of Base Formula

The change of base formula allows you to evaluate logarithms with any base using natural logarithms.

• Formula: $\log_b a = \frac{\ln a}{\ln b}$

• Example: $\log_6 14 = \frac{\ln(14)}{\ln(6)} \approx 1.47299$

Properties of Logarithms

Logarithms can be expressed as sums or differences using their properties.

• $\log_b(xy) = \log_b x + \log_b y$

• $\log_b(x^k) = k \log_b x$

• $\ln\left(\frac{x}{y}\right) = \ln x - \ln y$

Simplifying Exponential Expressions

• Example: $e^{\ln(4x+1)} = 4x+1$

Solving Exponential Equations

• Example: Solve $e^x = 13$

Take the natural logarithm of both sides: $\ln(e^x) = \ln(13)$ $x = \ln(13) \approx 2.5649$

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations can be solved using substitution, elimination, or matrices.

• Substitution: Solve one equation for a variable and substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

• Matrix Method: Use an augmented matrix to represent the system and solve using row operations or a calculator.

Writing Systems from Augmented Matrices

Each row of an augmented matrix represents an equation in the system.

Matrix	Corresponding System
$\begin{bmatrix} 1 & 4 & 3 & | & 5 \\ 5 & 7 & 2 & | & 8 \\ 9 & 10 & 11 & | & 12 \end{bmatrix}$	$x + 4y + 3z = 5$ $5x + 7y + 2z = 8$ $9x + 10y + 11z = 12$

Matrix Scalar Multiplication

Multiplying a matrix by a scalar means multiplying every entry by that scalar.

• Example: Find $4A$ for $A = \begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix}$

$4A = 4 \times \begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix} = \begin{bmatrix} 20 & 40 \\ 60 & 80 \end{bmatrix}$

Additional info: These notes cover key topics from College Algebra, including polynomial and rational functions, exponential and logarithmic functions, systems of equations, and matrices. Each section provides definitions, examples, and essential formulas for exam preparation.