College Algebra Study Notes: Equations, Graphs, and Functions

Linear Equations

Definition and Properties

Linear equations are fundamental in algebra and represent equations of the first degree, meaning the highest power of the variable is one. - Definition: A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and x is the variable. - Example: $3x + 4 = 20$ $3x = 16$ $x = \frac{16}{3}$ - Solving Linear Equations: Isolate the variable using inverse operations.

Solving Linear Equations: Examples

- Example 1: $5 + x = 2x - 1$ $5 + 1 = 2x - x$ $6 = x$ - Example 2: $2(x + 5) - 4 = 3(x - 1)$ $2x + 10 - 4 = 3x - 3$ $2x + 6 = 3x - 3$ $6 + 3 = 3x - 2x$ $9 = x$

Graphs

Distance and Midpoint Formulas

Graphs are visual representations of equations and relationships between variables. The distance and midpoint formulas are essential for analyzing points in the coordinate plane. - Distance Formula: Provides a method for computing the distance between two points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$: $d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ - Example: Find the distance between $(4, 5)$ and $(3, 2)$: $d = \sqrt{(4 - 3)^2 + (5 - 2)^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$ - Midpoint Formula: The midpoint $M$ of the segment from $P_1$ to $P_2$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Intercepts of a Graph

Intercepts are points where a graph crosses the axes. - x-intercept: The x-coordinate where the graph crosses the x-axis. - y-intercept: The y-coordinate where the graph crosses the y-axis. - Example: For the graph shown, x-intercepts are $(-3, 0), (2, 0), (4, 0)$ and y-intercept is $(0, 3)$.

Finding Intercepts from Equations

- To find the x-intercept, set $y = 0$ and solve for $x$. - To find the y-intercept, set $x = 0$ and solve for $y$. - Example: $y = x^2 - 4$ x-intercepts: $x^2 - 4 = 0 \rightarrow x = \pm 2$ y-intercept: $y = 0^2 - 4 = -4$

Symmetry in Graphs

Symmetry helps identify properties of graphs. - Symmetry with respect to x-axis: If for every $(x, y)$, $(x, -y)$ is also on the graph. - Symmetry with respect to y-axis: If for every $(x, y)$, $(-x, y)$ is also on the graph. - Symmetry with respect to origin: If for every $(x, y)$, $(-x, -y)$ is also on the graph.

Testing for Symmetry

- Replace $y$ with $-y$ for x-axis symmetry, $x$ with $-x$ for y-axis symmetry, and both for origin symmetry. - Example: Test $4x^2 + 9y^2 = 36$ for symmetry.

Slope of a Line

The slope measures the steepness of a line. - Positive slope: Line rises from left to right. - Negative slope: Line falls from left to right. - Zero slope: Line is horizontal. - Undefined slope: Line is vertical. - Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Equations of Lines

- Vertical line: $x = a$ - Horizontal line: $y = b$ - Point-slope form: $y - y_1 = m(x - x_1)$

Slope-Intercept Form

- Equation: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

Functions and Their Graphs

Definition of Functions

Functions are a special type of relation where each input has exactly one output. - Domain: The set of all possible input values (x-values). - Range: The set of all possible output values (y-values). - Function: A relation from set X to set Y such that each element of X corresponds to exactly one element of Y.

Identifying Functions

- If each input has only one output, the relation is a function. - Example: Menu items and prices: Each item has one price, so this is a function.

Domain and Range from Graphs

- Example: $y = 2x - 5$ is a function. $x^2 + y^2 = 1$ is not a function.

Function Notation

- Notation: $f(x)$ represents the value of the function at $x$. - Example: $f(x) = 2x - 5$ $f(1) = 2(1) - 5 = -3$ $f(3) = 2(3) - 5 = 1$

Difference Quotient

The difference quotient is used to measure the average rate of change of a function. - Definition: $\frac{f(x+h) - f(x)}{h}$, $h \neq 0$ - Example: For $f(x) = 2x^2 - 3x$, $f(x+h) = 2(x+h)^2 - 3(x+h)$

Finding Domain of Functions

- The domain is the set of real numbers for which the function is defined. - Exclude values that make denominators zero or radicands negative (for even roots). - Example: $f(x) = \frac{3x}{x^2 - 2}$ Domain: $x^2 - 2 \neq 0 \rightarrow x \neq \pm \sqrt{2}$

Operations on Functions

Functions can be added, subtracted, multiplied, or divided. - Example: $f(x) = 5x + 6$, $g(x) = 3x - 8$ $(f+g)(x) = (5x + 6) + (3x - 8) = 8x - 2$

Graphs of Functions and Non-Functions

- Example: $y = x^2$ is a function; $x^2 + y^2 = 1$ is not a function.

Properties of Functions: Even and Odd

- Even Function: $f(-x) = f(x)$ for all $x$ in the domain. Graph is symmetric with respect to the y-axis. - Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. Graph is symmetric with respect to the origin.

Increasing, Decreasing, and Constant Functions

- Increasing: Function rises as $x$ increases. - Decreasing: Function falls as $x$ increases. - Constant: Function remains unchanged as $x$ increases.

Local Maxima and Minima

- Local maximum: Occurs where the graph changes from increasing to decreasing. - Local minimum: Occurs where the graph changes from decreasing to increasing.

Evaluating Functions

- Substitute values into the function to find outputs. - Example: $f(x) = 2x^2 - x - 1$ $f(2) = 2(2)^2 - 2 - 1 = 8 - 2 - 1 = 5$

Summary Table: Types of Symmetry

Type of Symmetry	Test	Graph Property
x-axis	Replace y with -y	Symmetric about x-axis
y-axis	Replace x with -x	Symmetric about y-axis
Origin	Replace x with -x and y with -y	Symmetric about origin

Summary Table: Forms of Linear Equations

Form	Equation	Description
Point-slope	$y - y_1 = m(x - x_1)$	Line with slope m through (x1, y1)
Slope-intercept	$y = mx + b$	Line with slope m and y-intercept b
Vertical	$x = a$	Vertical line at x = a
Horizontal	$y = b$	Horizontal line at y = b