Graphs and the Rectangular Coordinate System

Plotting Points and Identifying Coordinates

The rectangular (Cartesian) coordinate system allows us to represent points in a plane using ordered pairs (x, y). Each point corresponds to a unique location defined by its horizontal (x) and vertical (y) distances from the origin (0, 0).

• Plotting Points: To plot a point, move x units along the x-axis and y units along the y-axis.

• Identifying Coordinates: Given a point on the plane, its coordinates are written as (x, y).

• Example: The point (3, -2) is 3 units right and 2 units down from the origin.

Distance and Midpoint Formulas

To analyze the relationship between two points, we use the distance and midpoint formulas.

• Distance Formula: The distance between points and is given by:

• Midpoint Formula: The midpoint between and is:

• Example: For points (1, 4) and (5, 2): Distance: Midpoint:

Application Problems

• Word problems may involve finding distances, midpoints, or interpreting coordinates in real-world contexts (e.g., navigation, geometry).

Graphing Equations and Analyzing Graphs

Intercepts and Graphing Using Intercepts

Intercepts are points where a graph crosses the axes.

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

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

• Graphing: Plot the intercepts and additional points to sketch the curve.

• Example: For , x-intercept: ; y-intercept: .

Symmetry of Graphs

• y-axis symmetry: Replace x with -x; if the equation is unchanged, the graph is symmetric about the y-axis.

• x-axis symmetry: Replace y with -y; if unchanged, symmetric about the x-axis.

• Origin symmetry: Replace x with -x and y with -y; if unchanged, symmetric about the origin.

• Example: is symmetric about the y-axis.

Calculator Skills: ZERO and INTERSECT Functions

• ZERO Function: Used to find x-intercepts (roots) of a function on a graphing calculator.

• INTERSECT Function: Used to approximate solutions to equations by finding where two graphs intersect.

• Application: Useful for solving equations graphically and checking solutions.

Linear Equations and Their Graphs

Slope of a Line

The slope measures the steepness of a line.

• Formula: For points and :

• Interpretation: Positive slope rises left to right; negative slope falls left to right.

• Example: Points (2, 3) and (5, 11):

Equations of Lines

• Point-Slope Form:

• Slope-Intercept Form:

• General Form:

• Finding the Equation: Use given information (point, slope, or two points) to write the equation in the desired form.

• Example: Through (1, 2) with slope 3:

Graphing Lines

• Using Slope and a Point: Plot the point, use the slope to find another point, draw the line.

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

Parallel and Perpendicular Lines

• Parallel Lines: Have equal slopes ().

• Perpendicular Lines: Slopes are negative reciprocals ().

• Example: is parallel to ; perpendicular to .

Word Problems Involving Lines

• Translate real-world situations into linear equations and solve for unknowns.

Circles in the Coordinate Plane

Center and Radius of a Circle

A circle is the set of all points equidistant from a fixed point (the center).

• Standard Form: where (h, k) is the center and r is the radius.

• General Form:

• Finding Center and Radius: Complete the square to convert general form to standard form.

• Example: has center (2, -3) and radius 4.

Completing the Square

• Used to rewrite the general form of a circle into standard form.

• Steps: Group x and y terms, complete the square for each variable, adjust the equation accordingly.

• Example: Complete the square: Center: (2, -3), Radius: 5

Word Problems Involving Circles

• Apply circle equations to real-world contexts, such as navigation, design, or geometry.

Functions and Their Properties

Relations and Functions

A relation is a set of ordered pairs. A function is a relation in which each input (x) has exactly one output (y).

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

• Example: is a function; is not.

Determining if an Equation is a Function

• For each x, there must be only one y.

• Check by solving for y in terms of x and seeing if multiple y-values result for a single x.

Evaluating Functions and Finding Domains

• Evaluating: Substitute the input value into the function.

• Domain: The set of all permissible input values (x-values) for which the function is defined.

• Example: For , domain is all real numbers except .

Operations with Functions

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

• Domain of Combined Functions: Intersection of the domains of the original functions, with additional restrictions for division.

Difference Quotient

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

• Formula: ,

• Example: For ,

Word Problems Involving Functions

• Apply function concepts to real-world scenarios, such as cost, revenue, or motion problems.

Analyzing Graphs of Functions

Graph Information: Domain, Range, Intercepts

• Domain: All x-values for which the function is defined.

• Range: All y-values the function attains.

• Intercepts: Points where the graph crosses the axes.

Even and Odd Functions

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

• Odd Function: for all x in the domain; symmetric about the origin.

• Neither: If neither condition holds.

• Example: is even; is odd; is neither.

Relative/Local Maxima and Minima; Increasing/Decreasing Intervals

• Relative Maximum: A point where the function changes from increasing to decreasing.

• Relative Minimum: A point where the function changes from decreasing to increasing.

• Increasing Interval: Where the function rises as x increases.

• Decreasing Interval: Where the function falls as x increases.

• Graphing Utility: Use calculators to approximate these values and intervals.

Average Rate of Change

• The average rate of change of from to is:

• Example: For from to :

Secant Line

• The secant line passes through two points on the graph of a function.

• Equation: Use the two points and to find the slope and write the equation in point-slope form.

Evaluating Difference Quotients

• Plug values into the difference quotient formula to compute the result.