Graphs and Functions: Precalculus Study Notes (Ch. 2)

Graphs and Functions

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is the foundation for graphing equations and functions in two dimensions. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0).

• Ordered pairs: Points are represented as (x, y).

• Quadrants: The plane is divided into four quadrants, numbered counterclockwise starting from the upper right.

• Plotting points: The x-value determines horizontal position; the y-value determines vertical position.

Example: Plot the points (4, 3), (-2, 2), (-2, -3), (0, 5), (-4, 0) on the graph.

Equations of Two Variables

Many equations in precalculus involve two variables, typically x and y. The solutions to these equations are ordered pairs (x, y) that satisfy the equation and can be graphed as points on the plane.

• Equations with one variable: Solutions are points on a line.

• Equations with two variables: Solutions are points on a plane.

Example: Graph the equation y = x + 2 by plotting points for several x-values.

Graphing Equations by Plotting Points

To graph an equation, select values for x, solve for y, and plot the resulting ordered pairs.

• Isolate y (or x) in the equation.

• Choose several values for x (or y).

• Calculate corresponding y (or x) values.

• Plot the points and connect with a line or curve.

Example: Graph the equation -2x + y = -1 by creating ordered pairs for x = -2, -1, 0, 1, 2.

Intercepts

Intercepts are points where a graph crosses the axes.

• x-intercept: The point where the graph crosses the x-axis (y = 0).

• y-intercept: The point where the graph crosses the y-axis (x = 0).

Example: Find the x- and y-intercepts of the equation 2x + 3y = 6.

Lines and Slope

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y to the change in x between two points.

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Example: Find the slope of the line passing through (2, 4) and (1, 2).

Forms of Linear Equations

Linear equations can be written in several forms:

Form	Equation	Description
Slope-Intercept	$y = mx + b$	m = slope, b = y-intercept
Point-Slope	$y - y_1 = m(x - x_1)$	m = slope, $(x_1, y_1)$ = point on line
Standard	$Ax + By = C$	A, B, C are constants

Example: Write the equation of a line passing through (1, -2) with slope 3 in point-slope form.

Parallel and Perpendicular Lines

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: $m_1 = m_2$

• Perpendicular: $m_1 \cdot m_2 = -1$

Example: Write the equation of a line parallel to y = 2x + 3 passing through (4, 1).

Relations and Functions

A relation is a set of ordered pairs. A function is a relation in which each input (x-value) corresponds to exactly one output (y-value).

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

Example: Determine if the graph of y = x^2 is a function.

Domain and Range

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

• Interval notation: Used to express domain and range, e.g., $[-2, 5)$.

• Set-builder notation: Describes the set using conditions, e.g., $\{x | x > 0\}$.

Example: Find the domain and range of $f(x) = \sqrt{x}$.

Finding the Domain of an Equation

Domain restrictions occur when certain x-values make the function undefined, such as taking the square root of a negative number or dividing by zero.

• Square root: $f(x) = \sqrt{x}$, domain is $x \geq 0$.

• Denominator: $f(x) = \frac{1}{x-2}$, domain is $x \neq 2$.

Example: Find the domain of $f(x) = \sqrt{x-3}$.

Graphs of Common Functions

Several basic functions frequently appear in precalculus. Their graphs have distinct shapes and properties.

Function	Equation	Domain	Range
Constant	$f(x) = c$	All real x	$y = c$
Identity	$f(x) = x$	All real x	All real y
Square	$f(x) = x^2$	All real x	$y \geq 0$
Cube	$f(x) = x^3$	All real x	All real y
Square Root	$f(x) = \sqrt{x}$	$x \geq 0$	$y \geq 0$
Cube Root	$f(x) = \sqrt[3]{x}$	All real x	All real y

Transformations of Functions

Transformations change the position or shape of a function's graph. Types include reflections, shifts, and stretches/compressions.

• Reflection: Flips the graph over an axis. $f(-x)$ reflects over the y-axis; $-f(x)$ reflects over the x-axis.

• Vertical shift: $f(x) + k$ shifts up/down by k units.

• Horizontal shift: $f(x - h)$ shifts right by h units; $f(x + h)$ shifts left by h units.

• Vertical stretch/compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Horizontal stretch/compression: $f(bx)$ compresses if $|b| > 1$, stretches if $0 < |b| < 1$.

Example: $g(x) = -f(x) + 2$ is a reflection over the x-axis and a shift up by 2 units.

Function Operations

Functions can be added, subtracted, multiplied, or divided. The domain of the resulting function is the intersection of the domains of the original functions, with additional restrictions for division.

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

Example: Given $f(x) = x^2$ and $g(x) = x + 1$, find $(f + g)(x)$ and $(f \cdot g)(x)$.

Function Composition

Function composition involves substituting one function into another. The notation $(f \circ g)(x)$ means $f(g(x))$.

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

• Domain: Values of x for which g(x) is in the domain of f.

Example: If $f(x) = x^2$ and $g(x) = x + 3$, then $(f \circ g)(x) = f(g(x)) = (x + 3)^2$.

Evaluating Compound Functions

To evaluate a compound function at a specific value, substitute the value into the inner function, then use the result as input for the outer function.

• Method 1: Compose, then evaluate.

• Method 2: Evaluate inside, then outside.

Example: For $f(x) = x^2$ and $g(x) = x + 1$, find $f(g(2))$. *Additional info: These notes cover the essential concepts of Chapter 2 in a Precalculus course, including coordinate systems, equations, lines, functions, domains, ranges, transformations, and function operations. The tables and examples have been expanded for clarity and completeness.*