Precalculus Study Guide: Functions, Equations, and Inequalities

Functions and Their Graphs

Linear, Quadratic, and Absolute Value Functions

Functions are mathematical relationships that assign each input exactly one output. Their graphs provide visual representations of these relationships.

• Linear Functions: The general form is $y = mx + b$, where m is the slope and b is the y-intercept.

• Quadratic Functions: The general form is $y = ax^2 + bx + c$. The graph is a parabola.

• Absolute Value Functions: The general form is $y = |x|$. The graph is V-shaped.

• Transformations: Shifting, stretching, or reflecting graphs by changing parameters.

Example: The graph of $y = x^2 + 4$ is a parabola shifted up by 4 units.

Intercepts and Graph Analysis

Finding x- and y-intercepts

Intercepts are points where the graph crosses the axes.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Example: For $y = 4x - 8$, the y-intercept is $-8$ and the x-intercept is $2$.

Solving Linear Equations

Methods and Solutions

Linear equations can be solved by isolating the variable.

• Standard form: $ax + b = c$

• Solution: $x = \frac{c - b}{a}$

Example: Solve $4(x - 6) = 18$:

• Expand: $4x - 24 = 18$

• Add 24: $4x = 42$

• Divide by 4: $x = 10.5$

Systems of Equations

Solving for Multiple Variables

Systems of equations involve finding values that satisfy all equations simultaneously.

• Substitution and elimination are common methods.

• Solutions may be unique, infinite, or nonexistent.

Example: Solve $y_1 = 8x + 44$, $y_2 = 3x - 6$, $y_1 = y_2$:

• Set $8x + 44 = 3x - 6$

• $5x = -50$

• $x = -10$

Formulas and Solving for Variables

Manipulating Formulas

Solving for a specified variable involves algebraic manipulation.

• Isolate the desired variable using inverse operations.

Example: Solve $A = \frac{2A - h}{h}$ for $A$.

Factoring and Expanding Expressions

Products and Standard Form

Factoring and expanding are used to simplify expressions and solve equations.

• Product: $(5x + 1)(5x^2 - 5)$

• Standard form: Expand and combine like terms.

Example: $(5x + 1)(5x^2 - 5) = 25x^3 + 5x^2 - 25x - 5$

Rational Expressions

Division and Simplification

Dividing rational expressions involves multiplying by the reciprocal.

• Example: $\frac{4}{3} \div \frac{8}{25} = \frac{4}{3} \times \frac{25}{8} = \frac{100}{24} = \frac{25}{6}$

Quadratic Equations

Solving by Formula and Completing the Square

Quadratic equations are solved using the quadratic formula or by completing the square.

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

• Completing the square: Rearranging to form $(x + p)^2 = q$

Example: Solve $x^2 + 10x - 24 = 0$ using the quadratic formula.

Discriminant and Nature of Solutions

Types of Roots

The discriminant $D = b^2 - 4ac$ determines the nature of the solutions of a quadratic equation.

• If $D > 0$: Two distinct real solutions

• If $D = 0$: One real solution

• If $D < 0$: Two complex solutions

Polynomial Equations and Zero Product Principle

Factoring and Solving

Polynomial equations can be solved by factoring and applying the zero product principle.

• Zero product principle: If $ab = 0$, then $a = 0$ or $b = 0$.

Example: $x^3 - 8x^2 + 15x = 0$ factors to $x(x^2 - 8x + 15) = 0$.

Inequalities and Interval Notation

Solving and Graphing

Inequalities are solved similarly to equations, but solutions are expressed as intervals.

• Interval notation: $(a, b)$ for all $x$ such that $a < x < b$

• Graphing: Use number lines to represent solution sets

Example: Solve $x - 3 \leq 5$; solution is $x \leq 8$, or $(-\infty, 8]$.

Absolute Value Inequalities

Solving and Graphing

Absolute value inequalities are solved by considering both positive and negative cases.

• General form: $|x - a| < b$ leads to $a - b < x < a + b$

Example: $|x - 3| < 3$ gives $0 < x < 6$

Applications: Word Problems

Translating and Solving

Word problems require translating real-world scenarios into mathematical equations.

• Identify variables and relationships

• Set up equations and solve

Example: A fax machine charges $2.50 for the first page and $0.65 for each subsequent page. The total price for$n$pages is$P = 0.65(n - 1) + 2.50$.

Summary Table: Types of Equations and Solution Methods

Equation Type	General Form	Solution Method
Linear	$ax + b = c$	Isolate variable
Quadratic	$ax^2 + bx + c = 0$	Factoring, quadratic formula, completing the square
Polynomial	$a_nx^n + ... + a_0 = 0$	Factoring, synthetic division
Inequality	$ax + b < c$	Isolate variable, interval notation
Absolute Value	$|x - a| < b$	Split into two inequalities

Additional info: Some context and explanations have been expanded for clarity and completeness, including general forms and solution steps for equations and inequalities.