Functions and Graphs

Section 1.1: Graphs and Graphing Utilities

This section introduces the rectangular coordinate system, plotting points, graphing equations, and interpreting graphs using graphing utilities.

• 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 position; the y-coordinate indicates vertical position.

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

• Graphing Utilities: Tools such as graphing calculators and software allow for visualization of equations. The viewing rectangle sets the visible range for 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 represent real-world data, such as divorce rates over time.

Section 1.2: Basics of Functions and Their Graphs

This section covers the definition of relations and functions, function notation, evaluating functions, and graphical analysis.

• 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.

Section 1.3: More on Functions and Their Graphs

This section discusses intervals of increase/decrease, relative extrema, symmetry, even/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 y-axis.

• Odd Function: ; symmetric about origin.

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

• Difference Quotient: for .

Section 1.4: Linear Functions and Slope

This section covers slope, forms of linear equations, graphing lines, and applications.

• 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.

• Applications: Linear models for real-world data.

Section 1.5: More on Slope

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

• Parallel Lines: Same slope.

• Perpendicular Lines: Product of slopes is .

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

• Average Rate of Change:

Section 1.6: Transformations of Functions

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

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

• Vertical Shift: shifts up; shifts down.

• Horizontal Shift: shifts left; shifts right.

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

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

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

• Sequence of Transformations: Apply shifts, stretches, and reflections in order.

Section 1.7: Combinations of Functions; Composite Functions

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

• Domain: Largest set of real numbers for which the function is defined.

• Algebra of Functions: , , , (domain is intersection of domains, except for division by zero).

• Composite Function:

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

• Writing Functions as Compositions: Express a function as a composition of two or more functions.

Section 1.8: Inverse Functions

This section covers verifying and finding inverse functions, the horizontal line test, and graphing inverses.

• Inverse Function: satisfies and .

• Finding Inverse: Replace with y, interchange x and y, solve for y, and relabel as .

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

• Graph of Inverse: Reflection of the original graph about the line .

Section 1.9: Distance and Midpoint Formulas; Circles

This section introduces formulas for distance and midpoint between points, and equations for circles.

• Distance Formula:

• Midpoint Formula:

• Circle: Set of points equidistant from a center.

• Standard Form of Circle:

• General Form:

• Converting to Standard Form: Complete the square for x and y terms.

Section 1.10: Modeling with Functions

This section demonstrates constructing functions from verbal descriptions and formulas to model real-world situations.

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

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

• Modeling with Geometric Formulas: Use formulas to express volume or area as functions of dimensions.

• Domain Considerations: Ensure variables represent meaningful values (e.g., positive lengths).