Functions and Graphs

Section 1.1: Graphs and Graphing Utilities

This section introduces the rectangular coordinate system and the basics of plotting points and graphing equations. It also covers the use of graphing utilities and the interpretation of graphs.

• Rectangular Coordinate System: Consists of a horizontal x-axis and a vertical y-axis intersecting at the origin (0,0).

• Plotting Points: Each point is represented by an ordered pair (x, y). The x-coordinate indicates horizontal movement, and the y-coordinate indicates vertical movement from the origin.

• Graphing Equations: An equation in two variables (x and y) can be graphed by plotting points that satisfy the equation.

• Graphing Utilities: Devices or software that allow for graphing equations by setting a viewing rectangle (range of x and y values).

• Intercepts: x-intercept is where the graph crosses the x-axis (y=0); y-intercept is where the graph crosses the y-axis (x=0).

• Interpreting Graphs: Graphs can be used to extract information, such as intercepts or values for real-world applications.

Example:

To plot the point (-2, 4), move 2 units left and 4 units up from the origin.

Section 1.2: Basics of Functions and Their Graphs

This section defines relations and functions, explains function notation, and describes how to graph functions and determine their domain and range.

• Relation: Any set of ordered pairs. Domain is the set of first components; Range is the set of second components.

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

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

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

• Domain and Range from Graphs: Domain is all x-values with points on the graph; range is all y-values.

• Intercepts from Graphs: x-intercepts are where the graph crosses the x-axis; y-intercept is where it crosses the y-axis.

Example:

Given the relation {(1,2), (2,3), (3,4)}, the domain is {1,2,3}, and the range is {2,3,4}.

Section 1.3: More on Functions and Their Graphs

This section discusses intervals of increase, decrease, and constancy, relative maxima and minima, symmetry, even and odd functions, piecewise functions, and the difference quotient.

• Increasing/Decreasing/Constant: A function is increasing if for , decreasing if , and constant if .

• Relative Maximum/Minimum: A relative maximum is a point higher than nearby points; a relative minimum is lower.

• Symmetry: Test for symmetry about the y-axis, x-axis, or origin by substituting or .

• Even Function: ; symmetric about the y-axis.

• Odd Function: ; symmetric about the origin.

• Piecewise Function: Defined by different expressions over different intervals.

• Difference Quotient: for .

Example:

For , , so f is even.

Section 1.4: Linear Functions and Slope

This section covers the definition of slope, forms of linear equations, and graphing lines using slope and intercepts.

• Slope:

• Point-Slope Form:

• Slope-Intercept Form:

• Horizontal Line: (slope = 0)

• Vertical Line: (slope undefined)

• General Form:

• Graphing Using Intercepts: Find x- and y-intercepts, plot, and draw the line.

Example:

Find the slope between (1,2) and (3,6): .

Section 1.5: More on Slope

This section explains slopes of parallel and perpendicular lines, slope as rate of change, and average rate of change.

• Parallel Lines: Same slope.

• Perpendicular Lines: Product of slopes is -1.

• Slope as Rate of Change: Describes how y changes with respect to x.

• Average Rate of Change:

Example:

For , average rate of change from x=1 to x=3: .

Section 1.6: Transformations of Functions

This section covers common functions and their graphs, and describes vertical/horizontal shifts, reflections, and stretching/shrinking.

• Common Functions: Constant, identity, absolute value, quadratic, square root, cubic, cube root.

• Vertical Shift: shifts up; shifts down.

• Horizontal Shift: shifts left; shifts right.

• Reflection: reflects about x-axis; reflects about y-axis.

• Vertical Stretch/Shrink: stretches if , shrinks if .

• Horizontal Stretch/Shrink: shrinks if , stretches if .

Example:

; shifts right 2 units.

Section 1.7: Combinations of Functions; Composite Functions

This section discusses combining functions (sum, difference, product, quotient) and forming composite functions.

• Domain: Exclude values causing division by zero or negative even roots.

• Algebra of Functions: , , ,

• Composite Function:

• Domain of Composite: x must be in domain of g, and g(x) in domain of f.

Example:

Let , . .

Section 1.8: Inverse Functions

This section defines inverse functions, explains how to find them, and describes the horizontal line test.

• Inverse Function: satisfies and .

• Finding Inverse: Replace f(x) with y, interchange x and y, solve for y.

• Horizontal Line Test: If no horizontal line crosses the graph more than once, f has an inverse.

• Graph of Inverse: Reflection of f about the line .

Example:

For , inverse is .

Section 1.9: Distance and Midpoint Formulas; Circles

This section provides formulas for the distance between two points, the midpoint of a segment, and the equation of a circle.

• Distance Formula:

• Midpoint Formula:

• Circle: Set of points equidistant from a center.

• Standard Form:

• General Form:

Example:

Circle with center (3,4) and radius 5: .

Section 1.10: Modeling with Functions

This section demonstrates constructing functions from verbal descriptions and formulas for real-world modeling.

• Modeling Costs: Express cost as a function of quantity, e.g., .

• Modeling Revenue: Revenue as a function of price and number of customers.

• Modeling with Geometry: Express volume or area as a function of dimensions.

Example:

For a box made from a 15x8 sheet with squares of side x cut from corners: .

Relevant Image

The following image is the cover of the textbook and is directly relevant as a visual identifier for the course materials: