Graphs, Functions, and Linear Equations: Foundations of Precalculus

1.1 Graphs and Graphing Utilities

Rectangular Coordinate System

The rectangular (Cartesian) coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at the origin (0,0). Each point in the plane is represented by an ordered pair (x, y).

• x-axis: Horizontal axis

• y-axis: Vertical axis

• Origin: The point (0, 0) where the axes intersect

• Quadrants: The plane is divided into four quadrants, labeled I (top right), II (top left), III (bottom left), IV (bottom right)

• Ordered Pair (x, y): Represents the location of a point

Plotting Points

To plot a point, move x units along the x-axis and y units along the y-axis from the origin.

• Example: The point (4, 5) is 4 units right and 5 units up from the origin.

Completing Tables and Identifying Quadrants

Given a set of points, you can determine their coordinates and which quadrant or axis they lie on.

Point	Coordinate	Quadrant
K	(-3, 5)	II
M	(0, -3)	y-axis
G	(0, 0)	Origin

Graphing Equations Using the Point-Plotting Method

To graph an equation, select values for x, compute the corresponding y values, plot the points, and connect them smoothly.

• Example: For $y = 4 - x$, choose x-values, compute y, plot, and connect.

Intercepts

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

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

Interpreting Graphs

Graphs can be used to interpret real-world data, such as rates or trends. For example, a linear model can represent the percentage of marriages ending in divorce over time.

1.2 Basic Functions and Their Graphs

Definitions

• Relation: Any set of ordered pairs. The set of all first components is the domain; the set of all second components is the range.

• Function: A relation in which each element of the domain corresponds to exactly one element of the range.

• Function Notation: $f(x)$ denotes the value of the function f at x.

Determining Functions

• If an equation is solved for y and gives more than one value of y for a given x, it is not a function of x.

• Example: $x^2 + y^2 = 1$ is not a function of x because for some x, there are two y-values.

Evaluating Functions

• To evaluate $f(x)$, substitute the given value for x and simplify.

• Example: If $f(x) = x^2 - 2x + 7$, then $f(-5) = (-5)^2 - 2(-5) + 7 = 25 + 10 + 7 = 42$.

Graphs of Functions

• Graphing functions helps visualize their behavior and relationships.

• Comparing $f(x)$ and $g(x)$ can show transformations such as shifts or stretches.

The Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Domain, Range, and Intercepts from a Graph

• Domain: All x-values for which the function is defined (look at the horizontal extent of the graph).

• Range: All y-values the function attains (vertical extent of the graph).

• Intercepts: Points where the graph crosses the axes.

1.3 More on Functions and Their Graphs

Increasing, Decreasing, and Constant Functions

• Increasing: $f(x_1) < f(x_2)$ for $x_1 < x_2$ in an interval.

• Decreasing: $f(x_1) > f(x_2)$ for $x_1 < x_2$ in an interval.

• Constant: $f(x_1) = f(x_2)$ for all $x_1, x_2$ in an interval.

Relative Maximum and Minimum

• Relative Maximum: $f(a)$ is a relative maximum if $f(a) f(x)$ for all x near a.

• Relative Minimum: $f(b)$ is a relative minimum if $f(b) f(x)$ for all x near b.

Even and Odd Functions

• 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.

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain.

• Example: $C(t) = \begin{cases} 20 & 0 \leq t \leq 60 \\ 20 + 0.40(t-60) & t > 60 \end{cases}$

Difference Quotient

The difference quotient is used to compute the average rate of change of a function over an interval and is foundational for calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$, $h \neq 0$

1.4 Linear Functions and Slope

Definitions

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

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

• Slope-Intercept Form: $y = mx + b$

• General Form: $Ax + By + C = 0$

Types of Slope

Type	Description
Positive Slope	Line rises from left to right
Negative Slope	Line falls from left to right
Zero Slope	Horizontal line
Undefined Slope	Vertical line

Parallel and Perpendicular Lines

• Parallel lines have equal slopes.

• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$

Average Rate of Change

• The average rate of change of $f$ from $x_1$ to $x_2$ is $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

1.6 Transformation of Functions

Vertical and Horizontal Shifts

• Vertical Shift: $y = f(x) + c$ shifts the graph up by $c$ units; $y = f(x) - c$ shifts it down by $c$ units.

• Horizontal Shift: $y = f(x - c)$ shifts the graph right by $c$ units; $y = f(x + c)$ shifts it left by $c$ units.

Reflections

• About the x-axis: $y = -f(x)$

• About the y-axis: $y = f(-x)$

Vertical and Horizontal Stretching/Shrinking

• Vertical Stretch: $y = c f(x)$, $c > 1$ stretches vertically; $0 < c < 1$ shrinks vertically.

• Horizontal Stretch: $y = f(cx)$, $c > 1$ shrinks horizontally; $0 < c < 1$ stretches horizontally.

1.7 Combinations of Functions: Composite Functions

Combining Functions

• Sum: $(f + g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

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

Composite Functions

• The composition $f \circ g$ is defined as $(f \circ g)(x) = f(g(x))$

• The domain of $f \circ g$ is all x such that x is in the domain of g and $g(x)$ is in the domain of f.

Summary Table: Key Function Properties

Property	Definition	Test/Formula
Function	Each input has one output	Vertical Line Test
Even Function	$f(-x) = f(x)$	Symmetry about y-axis
Odd Function	$f(-x) = -f(x)$	Symmetry about origin
Slope	Rate of change of a line	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Average Rate of Change	Change in function over interval	$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Additional info: This summary covers the foundational concepts of graphing, functions, linear equations, and transformations, as presented in the provided notes. Examples and exercises are referenced but not fully worked out, as the focus is on definitions, properties, and methods essential for Precalculus students.