Graphs Chapter 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AAA A

Rectangular Coordinates; Graphing Utilities Section 1.1

Rectangular Coordinate System

Example. Problem: Plot the points (0,7), ({6,0), (6,4) and ({3,{5) Answer:

Rectangular Coordinate System The points on the axes are not considered to be in any quadrant Quadrant I x > 0, y > 0 Quadrant II x 0 Quadrant III x < 0, y < 0 Quadrant IV x > 0, y < 0

Distance Formula Theorem [Distance Formula] The distance between two points P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ), denoted by d(P 1, P 2 ), is

Distance Formula Example. Problem: Find the distance between the points (6,4) and ({3,{5). Answer:

Midpoint Formula Theorem [Midpoint Formula] The midpoint M = (x,y) of the line segment from P 1 = (x 1, y 1 ) to P 2 = (x 2, y 2 ) is

Midpoint Formula Example. Problem: Find the midpoint of the line segment between the points (6,4) and ({3,{5) Answer:

Key Points Rectangular Coordinate System Distance Formula Midpoint Formula

Graphs of Equations in Two Variables Section 1.2

Solutions of Equations Solutions of an equation: Points that make the equation true when we substitute the appropriate numbers for x and y Example. Problem: Do either of the points ({3,{10) or (2,4) satisfy the equation y = 3x { 1? Answer:

Graphs of Equations Graph of an equation: Set of points in plane whose coordinates (x, y) satisfy the equation To plot a graph: List some solutions Connect the points More sophisticated methods seen later

Graphs of Equations Example. Problem: Graph the equation y = 3x{1 Answer:

Graphs of Equations Example. Problem: Graph the equation y 2 = x Answer:

Intercepts Intercepts: Points where a graph crosses or touches the axes, if any x-intercepts: x-coordinates of intercepts y-intercepts: y-coordinates of intercepts May be any number of x- or y- intercepts

Intercepts Example. Problem: Find all intercepts of the graph Answer:

Intercepts Finding intercepts from an equation To find the x-intercepts of an equation, set y=0 and solve for x To find the y-intercepts of an equation, set x=0 and solve for y

Intercepts Example. Problem: Find the intercepts of the equation 4x y 2 = 100 Answer:

Symmetry Symmetry with respect to the x- axis: If (x,y) is on the graph, then so is (x, {y) Symmetry with respect to the y- axis: If (x,y) is on the graph, then so is ({x, y) Symmetry with respect to the origin: If (x,y) is on the graph, then so is ({x, {y)

Symmetry and Graphs x-axis symmetry means that the portion of the graph below the x-axis is a reflection of the portion above it

Symmetry and Graphs y-axis symmetry means that the portion of the graph to the left of the y-axis is a reflection of the portion to the right of it

Symmetry and Graphs Origin symmetry Reflection across one axis, then the other Projection along a line through origin so that distances from the origin are equal Rotation of 180 ± about the origin

Symmetry and Equations To test an equation for x-axis symmetry: Replace y by {y y-axis symmetry: Replace x by {x origin symmetry: Replace x by {x and y by {y In each case, if an equivalent equation results, the graph has the appropriate symmetry

Symmetry and Equations Example. Problem: Test the equation x 2 {4x + y 2 { 5 = 0 for symmetry Answer:

Important Equations y = x 2 x-intercept: x = 0 y-intercept: y = 0 Symmetry: y-axis only

Important Equations x = y 2 x-intercept: x = 0 y-intercept: y = 0 Symmetry: x-axis only

Important Equations x-intercept: x = 0 y-intercept: y = 0 Symmetry: None

Important Equations y=x 3 x-intercept: x = 0 y-intercept: y = 0 Symmetry: Origin only

Important Equations y = x-intercept: None y-intercept: None Symmetry: Origin only

Key Points Solutions of Equations Graphs of Equations Intercepts Symmetry Symmetry and Graphs Symmetry and Equations Important Equations

Solving Equations in One Variable Using a Graphing Utility Section 1.3

Using Zero or Root to Approximate Solutions Example. Problem: Find the solutions to the equation x 3 { 6x + 3 = 0. Approximate to two decimal places. Answer:

Use Intersect to Solve Equations Example. Problem: Find the solutions to the equation {x 4 + 3x 3 + 2x 2 = {2x + 1. Approximate to two decimal places. Answer:

Key Points Using Zero or Root to Approximate Solutions Use Intersect to Solve Equations

Lines Section 1.4

Slope of a Line P = (x 1, y 1 ) and Q = (x 2,y 2 ) two distinct points P and Q define a unique line L If x 1  x 2, L is nonvertical. Its slope is defined as x 1  x 2, L is vertical. Slope is undefined.

Slope of a Line

Interpretation of the slope of a nonvertical line Average rate of change of y with respect to x, as x changes from x 1 to x 2

Any two distinct points serve to compute the slope The slope from P to Q is the same as the slope from Q to P Slope of a Line

Example. Problem: Compute the slope of the line containing the points (7,3) and ({2,{2) Answer:

Slope of a Line Move from left to right Line slants upward if the slope is positive Line slants downward if slope is negative Line is horizontal if the slope is 0 Larger magnitudes correspond to steeper slopes

Slope of a Line

Example. Problem: Draw the graph of the line containing the point (1,5) with a slope of Solution:

Equations of Lines Theorem [Equation of a Vertical Line] A vertical line is given by an equation of the form x = a where a is the x-intercept

Equations of Lines Example. Problem: Find an equation of the vertical line passing through the point ({1, 2) Answer:

Equations of Lines Theorem. [Equation of a Horizontal Line] A horizontal line is given by an equation of the form y = b where b is the y-intercept

Equations of Lines Example. Problem: Find an equation of the horizontal line passing through the point ({1, 2) Answer:

Point-Slope Form of a Line Theorem. [Point-Slope Form of an Equation of a Line] An equation of a nonvertical line of slope m that contains the point (x 1, y 1 ) is y { y 1 = m(x { x 1 )

Point-Slope Form of a Line Example. Problem: Find an equation of the line with slope passing through the point ({1, 2) Answer:

Point-Slope Form of a Line Example. Problem: Find an equation of the line containing the points ({1, 2) and (5,3). Answer:

Slope-Intercept Form of a Line Theorem. [Slope-Intercept Form of an Equation of a Line] An equation of a nonvertical line L with of slope m and y-intercept b y = mx + b

Slope-Intercept Form of a Line Example. Problem: Find the slope-intercept form of the line in the graph Answer:

General Form of a Line General form of a line L: Ax + By = C A, B and C are real numbers, A and B not both 0. Any line, vertical or nonvertical, may be expressed in general form The general form is not unique Any equation which is equivalent to the general form of a line is called a linear equation

Parallel Lines Parallel Lines: Two lines which do not intersect Theorem. [Criterion for Parallel Lines] Two nonvertical lines are parallel if and only if their slopes are equal and they have different y-intercepts.

Parallel Lines Example. Problem: Find the line passing through the point (1, {2) which is parallel to the line y = 3x + 2 Answer:

Perpendicular Lines Perpendicular lines: Two lines that intersect at a right angle

Perpendicular Lines Theorem. [Criterion for Perpendicular Lines] Two nonvertical lines are perpendicular if and only if the product of their slopes is {1. The slopes of perpendicular lines are negative reciprocals of each other

Perpendicular Lines Example. Problem: Find the line passing through the point (1, {2) which is parallel to the line y = 3x + 2 Answer:

Key Points Slope of a Line Equations of Lines Point-Slope Form of a Line Slope-Intercept Form of a Line General Form of a Line Parallel Lines Perpendicular Lines

Circles Section 1.5

Circles Circle: Set of points in xy-plane that are a fixed distance r from a fixed point (h,k) r is the radius (h,k) is the center of the circle

Standard Form of a Circle Standard form of an equation of a circle with radius r and center (h, k) is (x{h) 2 + (y{k) 2 = r 2 Standard form of an equation centered at the origin with radius r is x 2 + y 2 = r 2

Standard Form of a Circle Example. Problem: Graph the equation (x{2) 2 + (y+4) 2 = 9 Answer:

Unit Circle Unit Circle: Radius r = 1 centered at the origin Has equation x 2 + y 2 = 1

General Form of a Circle General form of the equation of a circle x 2 + y 2 + ax + by + c = 0 if this equation has a circle for a graph If given a general form, complete the square to put it in standard form

General Form of a Circle Example. Problem: Find the center and radius of the circle with equation x 2 + y 2 + 6x { 2y + 6 = 0 Answer:

Key Points Circles Standard Form of a Circle Unit Circle General Form of a Circle