College Algebra Study Notes: Graphs, Functions, and Linear & Quadratic Functions

Graphs and Functions

Ordered Pairs and the Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points are represented as ordered pairs (x, y), where x is the horizontal position and y is the vertical position.

• Graphing Ordered Pairs: To plot (x, y), move x units along the x-axis and y units along the y-axis.

• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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

• Equations with x and y: These represent lines or curves on the coordinate plane.

Example:

Find the distance and midpoint between (2, 3) and (6, 7):

• Distance: $d = \sqrt{(6-2)^2 + (7-3)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

• Midpoint: $M = \left( \frac{2+6}{2}, \frac{3+7}{2} \right) = (4, 5)$

Relations and Functions

A relation is any set of ordered pairs. A function is a relation where each input (x-value) has exactly one output (y-value).

• Determining Functions: A relation is a function if no x-value is paired with more than one y-value.

• Vertical Line Test: If any vertical line crosses a graph more than once, it is not a function.

Example:

The set {(1, 2), (2, 3), (3, 4)} is a function. The set {(1, 2), (1, 3)} is not a function.

Function Notation

Functions are often written as $f(x)$, which means "the value of function f at x." For example, if $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$.

Domain and Range

The domain is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Finding Domain: Look for x-values that make the function undefined (e.g., division by zero, negative square roots).

• Finding Range: Consider the possible y-values the function can produce.

Example:

For $f(x) = \sqrt{x}$, the domain is $x \geq 0$ and the range is $y \geq 0$.

Increasing, Decreasing, and Constant Functions

A function is increasing on an interval if its output rises as x increases, decreasing if its output falls, and constant if its output stays the same.

• Increasing: $f(x_1) < f(x_2)$ for $x_1 < x_2$

• Decreasing: $f(x_1) > f(x_2)$ for $x_1 < x_2$

• Constant: $f(x_1) = f(x_2)$ for $x_1 < x_2$

Even and Odd Functions

Even functions satisfy $f(-x) = f(x)$ for all x in the domain (symmetric about the y-axis). Odd functions satisfy $f(-x) = -f(x)$ (symmetric about the origin).

• Even Example: $f(x) = x^2$

• Odd Example: $f(x) = x^3$

Operations with Functions

Evaluating and Combining Functions

Functions can be added, subtracted, multiplied, divided, or composed.

• Addition: $(f + g)(x) = f(x) + g(x)$

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

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

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

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

Example:

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

Linear Functions

Forms of Linear Equations

Linear functions graph as straight lines and can be written in several forms:

• Slope-Intercept Form: $y = mx + b$

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

• Standard Form: $Ax + By = C$

Example:

A line with slope 2 passing through (1, 3): $y - 3 = 2(x - 1)$

Types of Linear Graphs

• Vertical Line: $x = a$ (undefined slope)

• Horizontal Line: $y = b$ (slope = 0)

• Positive Slope: Line rises left to right

• Negative Slope: Line falls left to right

Basic Function Graphs

Common Parent Functions

• Quadratic (Squared): $f(x) = x^2$

• Cubic: $f(x) = x^3$

• Absolute Value: $f(x) = |x|$

• Square Root: $f(x) = \sqrt{x}$

• Piecewise Functions: Defined by different expressions for different intervals of x

Graphing Techniques

Transformations of Graphs

Graphs can be transformed by stretching, shrinking, reflecting, or translating.

• Vertical Stretch/Shrink: $y = a f(x)$ (stretch if $|a| > 1$, shrink if $0 < |a| < 1$)

• Horizontal Stretch/Shrink: $y = f(bx)$ (shrink if $|b| > 1$, stretch if $0 < |b| < 1$)

• Reflection: $y = -f(x)$ (over x-axis), $y = f(-x)$ (over y-axis)

• Translation: $y = f(x) + k$ (up/down), $y = f(x - h)$ (right/left)

Example:

$y = 2(x - 3)^2 + 1$ is a vertical stretch by 2, shifted right 3 units and up 1 unit.

Quadratic Functions

Key Features of Quadratic Functions

A quadratic function has the form $f(x) = ax^2 + bx + c$.

• Vertex: The highest or lowest point on the graph. Vertex formula: $x = -\frac{b}{2a}$

• Axis of Symmetry: Vertical line through the vertex: $x = -\frac{b}{2a}$

• Direction: Opens upward if $a > 0$, downward if $a < 0$

Example:

For $f(x) = 2x^2 - 4x + 1$, vertex at $x = -\frac{-4}{2 \cdot 2} = 1$, so vertex is at (1, $f(1)$).

Summary Table: Parent Functions and Their Properties

Function	Equation	Domain	Range	Even/Odd
Linear	$f(x) = x$	All real x	All real y	Odd
Quadratic	$f(x) = x^2$	All real x	$y \geq 0$	Even
Cubic	$f(x) = x^3$	All real x	All real y	Odd
Absolute Value	$f(x) = |x|$	All real x	$y \geq 0$	Even
Square Root	$f(x) = \sqrt{x}$	$x \geq 0$	$y \geq 0$	Neither

Additional info: These notes expand on the study guide topics with definitions, formulas, and examples for self-contained review.