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 shows horizontal position; the y-coordinate shows vertical position.

• 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 and adjusting the 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: Use graphs to extract information, such as intercepts or values for real-world applications.

Section 1.2: Basics of Functions and Their Graphs

This section covers the definition of relations and functions, function notation, evaluating functions, and identifying domain and range.

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

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

• Function Notation: represents 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 maxima/minima, symmetry, even/odd functions, piecewise functions, and the difference quotient.

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

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

• 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 equations over different intervals.

• Difference Quotient: for ; used in calculus for rates of change.

Section 1.4: Linear Functions and Slope

This section covers slope, forms of linear equations, graphing lines, and modeling with linear functions.

• Slope: ; measures steepness.

• Point-Slope Form:

• Slope-Intercept Form:

• Horizontal Line: ; slope is 0.

• Vertical Line: ; slope is undefined.

• General Form:

• Graphing Using Intercepts: Find where the line crosses axes and draw through those points.

• Modeling Data: Use linear functions to model real-world relationships.

Section 1.5: More on Slope

This section explains 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 one variable changes with respect to another.

• Average Rate of Change:

Section 1.6: Transformations of Functions

This section covers common functions and their graphs, and transformations such as shifts, reflections, and stretching/shrinking.

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

• Vertical Shifts: shifts up; shifts down.

• Horizontal Shifts: 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) and forming composite functions.

• Domain of a Function: Set of x-values for which the function is defined.

• Algebra of Functions: , , ,

• Composite Function:

• Domain of Composite: x must be in domain of g, and g(x) 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 the definition, verification, and finding of inverse functions, as well as the horizontal line test.

• Inverse Function: reverses the effect of f; and .

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

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function does not have 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:

• Convert General to Standard: Complete the square for x and y terms.

Section 1.10: Modeling with Functions

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

• Modeling: Translate real-world situations into mathematical functions.

• Cost Models: Express cost as a function of input (e.g., linear feet).

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

• Geometric Models: Use formulas to model volume, area, etc., as functions of dimensions.

Relevant Image

The following image is the cover of the Precalculus textbook, which is directly relevant as it identifies the source and context of the study notes: