College Algebra: Core Concepts and Functions Study Guide

Equations and Inequalities

Solving Equations

Equations are mathematical statements asserting the equality of two expressions. In College Algebra, students learn to solve various types of equations, including linear, quadratic, and those involving irrational expressions.

• Linear Equations: Equations of the form $ax + b = 0$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$.

• Irrational Equations: Equations involving roots, such as $\sqrt{x+2} = 5$.

• Applications: Use equations to solve real-world problems.

Example: Solve $2x + 3 = 7$.

Solution: $2x = 4 \Rightarrow x = 2$.

Solving Inequalities

Inequalities compare two expressions and use symbols such as $<$, $>$, $\leq$, or $\geq$. Techniques include solving linear and absolute value inequalities.

• Linear Inequalities: $ax + b > 0$

• Absolute Value Inequalities: $|x| < a$

Example: Solve $|x - 3| < 5$.

Solution: $-5 < x - 3 < 5 \Rightarrow -2 < x < 8$.

Relations and Circles

Distance and Midpoint Formulas

These formulas are used to calculate the distance between two points and the midpoint of a segment in the coordinate plane.

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Circle Equations

The equation of a circle in the coordinate plane can be written in standard or general form.

• Standard (Center-Radius) Form: $(x - h)^2 + (y - k)^2 = r^2$

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$

• Completing the square can convert the general form to the center-radius form.

Example: Find the center and radius of $x^2 + y^2 - 4x + 6y - 12 = 0$.

Solution: Complete the square to rewrite in center-radius form.

Functions

Key Definitions

A function is a relation that assigns each input exactly one output. Functions are described by their domain, range, and rules of correspondence.

• Domain: Set of all possible input values.

• Range: Set of all possible output values.

• Independent Variable: Usually $x$.

• Dependent Variable: Usually $y$ or $f(x)$.

Graphical Features

Understanding the graph of a function helps identify its behavior, including intervals of increase/decrease and points of maximum/minimum.

• Increasing/Decreasing: Where the function rises or falls as $x$ increases.

• Constant: Where the function remains unchanged.

Example: The function $f(x) = x^2$ is decreasing for $x < 0$, increasing for $x > 0$.

Linear Functions

Linear functions have the form $f(x) = mx + b$ and graph as straight lines.

• Slope: Measures steepness, $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Intercepts: Points where the graph crosses axes.

• Forms: Point-slope and slope-intercept forms.

Example: $y = 2x + 3$ has slope $2$and$y$-intercept $3$.

Equations of Lines and Properties

Forms of Linear Equations

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

Lines can be parallel (same slope) or perpendicular (slopes are negative reciprocals).

Basic Functions

Common Functions

Several basic functions are foundational in algebra:

• Constant Function: $f(x) = c$

• Identity Function: $f(x) = x$

• Absolute Value Function: $f(x) = |x|$

• Piecewise Functions: Defined by different expressions over different intervals.

Example: $f(x) = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

Graphs of Functions

Transformations

Transformations alter the graph of a function in predictable ways.

• Vertical Shifting: $f(x) + k$ shifts up/down.

• Horizontal Shifting: $f(x + h)$ shifts left/right.

• Vertical Stretching/Shrinking: $a f(x)$ stretches/shrinks vertically.

• Reflection: $-f(x)$ reflects over the $x$-axis.

Example: $f(x) = (x - 2)^2 + 3$ is shifted right by 2 and up by 3.

Properties of Graphs

Key properties include symmetry, intercepts, and intervals of increase/decrease.

• Symmetry: Even functions are symmetric about the $y$-axis; odd functions about the origin.

• Intercepts: Where the graph crosses axes.

Functions and Their Graphs

Domain and Range

Determining the domain and range is essential for understanding the behavior of functions.

• Domain: All $x$-values for which the function is defined.

• Range: All possible $f(x)$ values.

Example: For $f(x) = \sqrt{x}$, domain is $x \geq 0$.

Algebra of Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(f \cdot g)(x) = f(x)g(x)$

• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x + 1)^2$.

Effect on Domain

Operations on functions can affect the domain, especially division and composition.

• For $\frac{f}{g}$, exclude values where $g(x) = 0$.

• For $f(g(x))$, domain is restricted to values where $g(x)$ is in the domain of $f$.

Table: Types of Function Transformations

Transformation	Effect on Graph	Equation Form
Vertical Shift	Moves graph up/down	$f(x) + k$
Horizontal Shift	Moves graph left/right	$f(x + h)$
Vertical Stretch/Shrink	Stretches or shrinks graph vertically	$a f(x)$
Reflection	Reflects graph over axis	$-f(x)$ or $f(-x)$