College Algebra Final Exam Review: Comprehensive Study Notes

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving these equations involves isolating the variable on one side.

• Key Point: Use inverse operations to isolate the variable.

• Example: $9 - 2(3 - x) = 5x$ Expand and solve for $x$:

• $9 - 6 + 2x = 5x$

• $3 + 2x = 5x$

• $3 = 3x$

• $x = 1$

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: $x^2 - 5x + 6 = 0$ Factoring: $(x - 2)(x - 3) = 0$ Solutions: $x = 2$, $x = 3$

Solving Radical Equations

Radical equations contain variables within a root. Isolate the radical and then square both sides to eliminate the root, checking for extraneous solutions.

• Example: $\sqrt{x + 1} = x - 1$ Square both sides: $x + 1 = (x - 1)^2$

• Expand: $x + 1 = x^2 - 2x + 1$

• Rearrange: $x^2 - 3x = 0$

• Factor: $x(x - 3) = 0$ Solutions: $x = 0$, $x = 3$ (Check for extraneous solutions)

Solving Absolute Value Equations and Inequalities

Absolute value equations and inequalities require considering both the positive and negative cases.

• Example: $|x - 1| = 2$ Cases: $x - 1 = 2$ or $x - 1 = -2$ Solutions: $x = 3$, $x = -1$

• Graphing: Represent solutions on a number line.

Compound and Rational Inequalities

Compound inequalities involve two inequalities joined by 'and' or 'or'. Rational inequalities involve expressions with variables in the denominator.

• Example: $8 < 4x < 16$ Divide by 4: $2 < x < 4$

• Rational Inequality Example: $\frac{x-4}{x-1} \geq 0$ Find critical points and test intervals.

Functions and Graphs

Function Basics

A function is a relation in which each input has exactly one output. Functions can be represented by equations, tables, or graphs.

• Key Point: The domain is the set of all possible input values; the range is the set of all possible output values.

• Example: $f(x) = 2x + 1$ Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$

Graphing Functions

Graphing involves plotting points and understanding transformations such as shifts, reflections, and stretches.

• Transformation Example: $h(x) = -(x - 4)^2 + 7$ 1) Shift 4 units right 2) Reflect over x-axis 3) Shift 7 units up

Difference Quotient

The difference quotient is used to measure the average rate of change of a function.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = 2x + 8$, $\frac{f(x+h) - f(x)}{h} = \frac{2(x+h) + 8 - (2x + 8)}{h} = 2$

Function Composition

Function composition involves applying one function to the results of another.

• Notation: $(g \circ f)(x) = g(f(x))$

• Example: If $f(x) = x + 10$, $g(x) = 4x$, then $(g \circ f)(x) = 4(x + 10) = 4x + 40$

Polynomial and Rational Functions

Polynomial Functions

Polynomials are expressions consisting of variables and coefficients, involving only non-negative integer powers of variables.

• End Behavior: Determined by the leading term.

• Zeros and Multiplicity: The zeros of a polynomial are the values of $x$ where $f(x) = 0$. Multiplicity refers to how many times a zero occurs.

• Example: $f(x) = (x - 1)^2(x + 3)$ Zero at $x = 1$ (multiplicity 2, touches x-axis), $x = -3$ (multiplicity 1, crosses x-axis)

Graphing Polynomial Functions

Graphing involves plotting zeros, determining end behavior, and identifying turning points.

• Example: $f(x) = x^3 - 2x^2 - x + 2$ Find zeros and sketch the graph.

Rational Functions and Inequalities

Rational functions are quotients of polynomials. Their domain excludes values that make the denominator zero.

• Example: $f(x) = \frac{x^2 - 6}{x - 2}$ Domain: $x \neq 2$

• Solving Rational Inequalities: Set numerator and denominator equal to zero to find critical points, test intervals.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a \cdot b^x$, where $a$ is a constant and $b$ is the base.

• Growth and Decay: If $b > 1$, the function grows; if $0 < b < 1$, it decays.

• Example: Population growth: $P = P_0 e^{rt}$

Logarithmic Functions

Logarithms are the inverses of exponential functions. $y = \log_b x$ means $b^y = x$.

• Properties: $\log_b(xy) = \log_b x + \log_b y$ $\log_b(x^n) = n \log_b x$

• Solving Logarithmic Equations: Isolate the logarithm and rewrite in exponential form.

• Example: $\log_2(x - 1) = 3$ Rewrite: $x - 1 = 2^3 = 8$ Solution: $x = 9$

Systems of Equations and Inequalities

Solving Systems of Linear Equations

Systems of equations can be solved by substitution, elimination, or graphing.

• Example: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ Add equations: $2x = 6 \Rightarrow x = 3$ Substitute: $3 + y = 5 \Rightarrow y = 2$

Graphing Systems of Inequalities

Graph each inequality and shade the region representing the solution set.

• Example: $y > 2x - 4$ and $y < -x + 3$ Graph both lines and shade the overlapping region.

Additional Info

• Some problems involve interpreting graphs, identifying intervals of increase/decrease, and understanding function transformations.

• Problems also include finding equations of lines given points and slopes, and using interval notation for solution sets.

Table: Properties of Polynomial Zeros and Multiplicity

Zero	Multiplicity	Graph Behavior
$x = a$	Odd	Crosses x-axis
$x = a$	Even	Touches x-axis and turns around