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Functions and Graphs: Foundations, Properties, and Linear Models

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Functions and Graphs

Basics of Functions and Their Graphs

Understanding functions and their graphical representations is fundamental in algebra and trigonometry. This section introduces the concepts of relations, functions, and how to interpret their graphs.

  • Relation: Any set of ordered pairs. The domain is the set of all first components (inputs), and the range is the set of all second components (outputs).

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

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

  • Evaluating Functions: Substitute the input value into the function's formula to find the output.

  • Graph of a Function: The set of all ordered pairs (x, f(x)).

Example: The relation {(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (40, 21.2)} has domain {0, 10, 20, 30, 40} and range {9.1, 6.7, 10.7, 13.2, 21.2}.

Determining Functions from Relations and Equations

To determine if a relation or equation defines a function, check if each input has only one output.

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

Equation showing y^2 = 1 - x^2 and y = ±sqrt(1 - x^2)

Example: The equation gives , so for some x-values, there are two y-values. Thus, it does not define y as a function of x.

Graphing Functions and the Vertical Line Test

Functions can be graphed by plotting points. The vertical line test is used to determine if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

Graphs of two linear functions on the same axesGraph failing the vertical line test (not a function)Graph passing the vertical line test (function)

Obtaining Information from Graphs

Graphs provide valuable information about a function, such as the value at a specific input, domain, range, and intercepts.

  • Domain: All x-values for which the function is defined (visible on the x-axis).

  • Range: All y-values the function attains (visible on the y-axis).

  • Intercepts: Points where the graph crosses the axes. The x-intercept(s) occur where y = 0; the y-intercept occurs where x = 0.

Graph showing function values and interceptsSolution showing domain and range from a graphSolution showing domain and range from a graphGraph of a piecewise function with domain and rangeGraph showing x- and y-intercepts

More on Functions and Their Graphs

Increasing, Decreasing, and Constant Functions

A function's behavior can be described as increasing, decreasing, or constant over intervals.

  • Increasing: f(x) increases as x increases on an interval.

  • Decreasing: f(x) decreases as x increases on an interval.

  • Constant: f(x) remains the same as x increases on an interval.

Intervals of increase, decrease, and constancyGraph showing increasing, decreasing, and constant intervals

Relative Maxima and Minima

A function may have points where it reaches a local highest or lowest value, called relative maximum or relative minimum.

  • Relative Maximum: f(a) is a relative maximum if it is greater than all nearby values.

  • Relative Minimum: f(b) is a relative minimum if it is less than all nearby values.

Graph showing relative maximum and minimum

Even and Odd Functions

Functions can be classified by their symmetry:

  • Even Function: for all x in the domain. The graph is symmetric about the y-axis.

  • Odd Function: for all x in the domain. The graph is symmetric about the origin.

Solution showing a function is neither even nor odd

Piecewise Functions

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

  • To evaluate, determine which piece applies for the given input.

  • To graph, plot each piece over its specified interval.

Graph of a piecewise function

Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and is defined as:

Algebraic expansion for difference quotientSimplification of the difference quotient

Linear Functions and Slope

Definition and Calculation of Slope

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

Formula for slope between two pointsExample calculation of slope

Equations of Lines

  • Point-Slope Form:

  • Slope-Intercept Form:

  • General Form:

  • Horizontal Line: (slope = 0)

  • Vertical Line: (slope undefined)

Graphing Linear Equations

To graph :

  1. Plot the y-intercept (0, b).

  2. Use the slope (rise/run) to find a second point.

  3. Draw a line through the points.

Graphing using slope and y-interceptLine drawn through two points using slope and interceptGraph of a horizontal line y = 3

Using Intercepts to Graph

To graph :

  1. Find the x-intercept (set y = 0, solve for x).

  2. Find the y-intercept (set x = 0, solve for y).

  3. Draw a line through these intercepts.

Finding slope and y-intercept from general formGraph of a line using intercepts

Modeling Data with Linear Functions

Linear functions can be used to model real-world data and make predictions. The slope represents the rate of change, and the y-intercept represents the initial value.

Calculation of slope from data points

More on Slope

Parallel and Perpendicular Lines

  • Parallel Lines: Nonvertical lines are parallel if they have the same slope.

  • Perpendicular Lines: Nonvertical lines are perpendicular if the product of their slopes is -1.

Finding slope for perpendicular lines

Slope as Rate of Change

The slope of a linear function represents the rate of change of the dependent variable with respect to the independent variable.

Example: If the number of adult men living alone increased from 9 million in 1990 to 14.7 million in 2008, the slope is million per year.

Average Rate of Change of a Function

The average rate of change of a function between and is:

Calculation of average rate of change

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