Functions and Graphs: Core Concepts and Transformations in Precalculus

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Functions and Graphs

Relations, Functions, Domain, and Range

Understanding the distinction between relations and functions is foundational in precalculus. A relation is any set of ordered pairs, while a function is a relation in which each input (domain value) corresponds to exactly one output (range value).

Domain: The set of all possible input values (x-values) for a function.
Range: The set of all possible output values (y-values) for a function.
Set Notation: Domains and ranges are often expressed in set notation, e.g., {x | x ≥ 0}.

To determine if a relation is a function, check that no input value is paired with more than one output value.

Increasing, Decreasing, and Constant Intervals

The behavior of a function can be described by intervals where it is increasing, decreasing, or constant:

Increasing: A function f(x) is increasing on an interval if, as x increases, f(x) also increases.
Decreasing: A function f(x) is decreasing on an interval if, as x increases, f(x) decreases.
Constant: A function f(x) is constant on an interval if, as x increases, f(x) remains the same.

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry:

Even Function: f(x) is even if for all x in the domain. Even functions are symmetric about the y-axis.
Odd Function: f(x) is odd if for all x in the domain. Odd functions have rotational symmetry about the origin.
Neither: If a function does not satisfy either condition, it is neither even nor odd.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. To evaluate a piecewise function, determine which interval the input belongs to and use the corresponding expression.

Example:

Relative Maximum and Minimum

Relative (or local) maxima and minima are important for understanding the behavior of functions:

Relative Maximum: A function value f(a) is a relative maximum if it is greater than all nearby values.
Relative Minimum: A function value f(b) is a relative minimum if it is less than all nearby values.

Definitions of relative maximum and minimum with graph

Graphical Analysis of Functions

Graphs provide visual insight into the properties of functions, including intercepts, intervals of increase/decrease, and extrema.

x-intercept: The point(s) where the graph crosses the x-axis (set y = 0).
y-intercept: The point where the graph crosses the y-axis (set x = 0).
Relative Maxima/Minima: Highest/lowest points in a local region of the graph.

Graph of a function with local maxima and minima

Linear Functions and Slope

Slope and Rate of Change

The slope of a line measures its steepness and is calculated as:

Parallel Lines: Have the same slope.
Perpendicular Lines: Have slopes that are negative reciprocals; their product is -1.

Forms of Linear Equations

Standard Form:
Slope-Intercept Form:
Point-Slope Form:

Special cases include horizontal lines (, slope = 0) and vertical lines (, slope undefined).

Finding Intercepts

x-intercept: Set y = 0 and solve for x.
y-intercept: Set x = 0 and solve for y.

Average Rate of Change

The average rate of change of a function between two points is the slope of the secant line connecting those points:

Secant line representing average rate of change on a height vs. age graph

Difference Quotient

Definition and Application

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

, for

Used to find the slope of the secant line between two points on a function.

Transformations of Functions

Reflections

Reflection about the x-axis: reflects the graph over the x-axis.
Reflection about the y-axis: reflects the graph over the y-axis.

Reflection about the x-axis Reflection about the y-axis

Vertical and Horizontal Shifts

Vertical Shift Up: shifts the graph up by c units.
Vertical Shift Down: shifts the graph down by c units.
Horizontal Shift Left: shifts the graph left by c units.
Horizontal Shift Right: shifts the graph right by c units.

Vertical shifts of a function Horizontal shifts of a function

Vertical Stretching and Shrinking

Vertical Stretch: , stretches the graph vertically by a factor of c.
Vertical Shrink: , shrinks the graph vertically by a factor of c.

Vertical stretching and shrinking

Horizontal Stretching and Shrinking

Horizontal Shrink: , shrinks the graph horizontally by a factor of c.
Horizontal Stretch: , stretches the graph horizontally by a factor of c.

Horizontal stretching and shrinking

Summary Table of Transformations

The following table summarizes the main types of function transformations and their effects:

To Graph:	Draw the Graph of f and:	Changes in the Equation of y = f(x)
Vertical shifts	Raise/lower the graph by c units	c is added/subtracted to f(x)
Horizontal shifts	Shift left/right by c units	x is replaced with x + c or x - c
Reflection about the x-axis	Reflect over x-axis	f(x) is multiplied by -1
Reflection about the y-axis	Reflect over y-axis	x is replaced with -x
Vertical stretching/shrinking	Multiply y-coordinates by c	f(x) is multiplied by c
Horizontal stretching/shrinking	Divide x-coordinates by c	x is replaced with c x

Summary table of function transformations

Practice and Application

Apply these concepts to analyze, graph, and transform functions. Practice problems may involve identifying intervals of increase/decrease, finding intercepts, classifying functions as even/odd/neither, and describing or sketching transformations.