Digital Lesson on Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables.

Slides:



Advertisements
Similar presentations
Trigonometric Form of a Complex Number
Advertisements

Section P.1 – Graphs and Models. How to Graph xy Make a table to graph y = x Use your knowledge of relations and functions.
Advanced Algebra Trigonometry Appendix B
Graphing Equations: Point-Plotting, Intercepts, and Symmetry
Sullivan Algebra and Trigonometry: Section 2.2 Graphs of Equations Objectives Graph Equations by Plotting Points Find Intercepts from a Graph Find Intercepts.
Digital Lesson Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables x and.
Solving Two-Variable Systems of Linear Equations
Section 3.2 Graphs of Equations Objectives: Find the symmetries of equations with respect to x, y axis and origin. Use the graphical interpretation In.
Polar Coordinates and Graphs of Polar Equations Digital Lesson.
Solving Systems of Linear Equations Digital Lesson.
10.6 Equations of Circles Advanced Geometry. What do you need to find the equation of a circle? Coordinates of the Center of the circle. Radius – Distance.
Solving Equations. Is a statement that two algebraic expressions are equal. EXAMPLES 3x – 5 = 7, x 2 – x – 6 = 0, and 4x = 4 To solve a equation in x.
Graphs of Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of a function y = f (x) is a set of.
Polar Coordinates and Graphs of Polar Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The polar coordinate system is formed.
Copyright © Cengage Learning. All rights reserved. 1 Functions and Their Graphs.
coordinates, lines and increment
Copyright © Cengage Learning. All rights reserved.
Chapter 1 Functions and Their Graphs. 1.1 Rectangular Coordinates You will know how to plot points in the coordinate plane and use the Distance and Midpoint.
Preparation for Calculus P Copyright © Cengage Learning. All rights reserved.
OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Equations Sketch a graph by plotting points. Find the.
Graphs of Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of a function y = f (x) is a set of.
Example: The graph of x = | y | – 2 shown below, is symmetric to x-axis y x 1 2 –323 A graph is symmetric to x- axis if whenever (x, y) is on graph, so.
P.4 GRAPHS OF EQUATIONS Copyright © Cengage Learning. All rights reserved.
Sullivan Algebra and Trigonometry: Section 10.2 Objectives of this Section Graph and Identify Polar Equations by Converting to Rectangular Coordinates.
Graphs of Equations Objective: To use many methods to sketch the graphs of equations.
6-2 Conic Sections: Circles Geometric definition: A circle is formed by cutting a circular cone with a plane perpendicular to the symmetry axis of the.
Digital Lesson Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables x and.
Polar Coordinates and Graphs of Polar Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The polar coordinate system is formed.
9.7 Graphs of Polar Equations Digital Lesson. HWQ Convert the polar equation to rectangular form. Give the equation in standard form. Copyright © by Houghton.
Chapter 1 Functions and Their Graphs. Copyright © Houghton Mifflin Company. All rights reserved.1 | 2 Section 1.1: Figure 1.1, The Cartesian Plane.
Chapter 1.1 – Graphs of Equations What you should learn 1. Sketch graphs of equations 2. Find x- and y- intercepts of graphs of equations 3. Use symmetry.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. MATH 108 Section Coordinate Plane and Graphs.
2.1Intercepts;Symmetry;Graphing Key Equations
Copyright © Cengage Learning. All rights reserved.
Graphs of Equations in Two Variables
Find the missing coordinate in the ordered pair
Digital Lesson Graphs of Equations.
Warmup 1. Find the equation of a circle with center (-4,1) and radius 3 2. Find an equation of a circle with center at the origin passing through P(4,
Polar Coordinates and Graphs of Polar Equations
Solving Systems of Linear Equations
Copyright © Cengage Learning. All rights reserved.
Intercepts, Symmetry, Even/Odd and intersections
Graphs of Equations In Two Variables; Intercepts; Symmetry
Sullivan Algebra and Trigonometry: Section 2.2
Solving Systems of Linear Equations
1.2 Introduction to Graphing Equations
Linear Equations in Two Variables
Notes Over 1.1 To check for y-axis symmetry replace x with -x.
Polar Coordinates and Graphs of Polar Equations
Intercepts and Symmetry
Graphs of Equations Objectives: Find intercepts from a Graph
Analyzing Graphs of Functions and Relations Unit 1 Lesson 2
1.3 Symmetry; Graphing Key Equations; Circles
Graphs and Graphing Utilities
Digital Lesson Graphs of Equations.
Introduction to Graphing
Warm-up What is the distance formula between two points?
MATH1910 Chapter P Section 1 Graphs and Models.
Graphs of Equations Objectives: Find intercepts from a Graph
Linear Equations in Two Variables
Linear Equations in Two Variables
Graphing Key Equations
Polar Coordinates and Graphs of Polar Equations
9.7 Graphs of Polar Equations
P.3 Graphs of Equations (Part 4).
Digital Lesson Graphs of Equations.
SYSTEM OF LINEAR EQUATIONS
The Graph of an Equation Objective: 1. Sketch graphs of equations 2. Find x- and y-intercepts of graphs of equations 3. Find equations of and sketch graphs.
Solving Linear Systems by Graphing
Presentation transcript:

Digital Lesson on Graphs of Equations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation. For instance, the point (–1, 3) is on the graph of 2y – x = 7 because the equation is satisfied when –1 is substituted for x and 3 is substituted for y. That is, 2y – x = 7 Original Equation 2(3) – (–1) = 7 Substitute for x and y. 7 = 7 Equation is satisfied. Definition of Graph

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 To sketch the graph of an equation, 1.Find several solution points of the equation by substituting various values for x and solving the equation for y. 2. Plot the points in the coordinate plane. 3.Connect the points using straight lines or smooth curves. Sketching Graphs

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Sketch the graph of y = –2x Find several solution points of the equation. xy = –2x + 3(x, y) –2y = –2(–2) + 3 = 7(–2, 7) –1y = –2(–1) + 3 = 5(–1, 5) 0y = –2(0) + 3 = 3(0, 3) 1y = –2(1) + 3 = 1(1, 1) 2y = –2(2) + 3 = –1(2, –1) Example: Sketch Graph (Linear Function)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Sketch the graph of y = –2x Plot the points in the coordinate plane –4 x y xy(x, y) –27(–2, 7) –15(–1, 5) 03(0, 3) 11(1, 1) 2–1(2, –1) Example continued

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Sketch the graph of y = –2x Connect the points with a straight line –4 x y Example continued

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Sketch the graph of y = (x – 1) 2. xy(x, y) –29(–2, 9) –14(–1, 4) 01(0, 1) 10(1, 0) 21(2, 1) 34(3, 4) 49(4, 9) y x –2 Example: Sketch Graph (Quadratic Function)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Graph

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Sketch the graph of y = | x | + 1. xy(x, y) –23(–2, 3) –12(–1, 2) 01(0, 1) 12(1, 2) 23(2, 3) y x – Example: Sketch Graph (Absolute Value Function)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 The points at which the graph intersects the the x- or y-axis are called intercepts. A point at which the graph of an equation meets the y-axis is called a y-intercept. It is possible for a graph to have no intercepts, one intercept, or several intercepts. If (x, 0) satisfies an equation, then the point (x, 0) is called an x-intercept of the graph of the equation. If (0, y) satisfies an equation, then the point (0, y) is called a y-intercept of the graph of the equation. Definition of Intercepts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 To find the x-intercepts of the graph of an equation, substitute 0 for y in the equation and solve for x. To find the y-intercepts of the graph of an equation algebraically, substitute 0 for x in the equation and solve for y. Procedure for finding the x- and y- intercepts of the graph of an equation algebraically: Finding Intercepts Algebraically

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Find the x- and y-intercepts of the graph of y = x 2 + 4x – 5. To find the x-intercepts, let y = 0 and solve for x. 0 = x 2 + 4x – 5 Substitute 0 for y. 0 = (x – 1)(x + 5) Factor. x – 1 = 0 x + 5 = 0 Set each factor equal to 0. x = 1 x = –5 Solve for x. So, the x-intercepts are (1, 0) and (–5, 0). To find the y-intercept, let x = 0 and solve for y. y = (0) – 5 = –5 So, the y-intercept is (0, –5). Example: Find Intercepts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 To find the x-intercepts of the graph of an equation, locate the points at which the graph intersects the x-axis. Procedure for finding the x- and y-intercepts of the graph of an equation graphically: To find the y-intercepts of the graph of an equation, locate the points at which the graph intersects the y-axis. Finding Intercepts Graphically

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Find the x- and y-intercepts of the graph of x = | y | – 2 shown below. y x 1 2 –323 The x-intercept is (–2, 0). The y-intercepts are (0, 2) and (0, –2). The graph intersects the x-axis at (–2, 0). The graph intersects the y-axis at (0, 2) and at (0, –2). Example: Find Intercepts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Graphical Tests for Symmetry A graph is symmetric with respect to the y-axis if, whenever (x, y) is on the graph, (-x, y) is also on the graph. As an illustration of this we graph y = x 2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Graphical Tests for Symmetry A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph, (x, -y) is also on the graph. As an illustration of this we graph y 2 = x.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Graphical Tests for Symmetry A graph is symmetric with respect to the origin if, whenever (x, y) is on the graph, (-x, -y) is also on the graph. As an illustration of this we graph y = x 3

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Algebraic Tests for Symmetry The graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields an equivalent equation. The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation. The graph of an equation is symmetric with respect to the origin if replacing x with –x and replacing y with –y yields an equivalent equation. The algebraic tests for symmetry are as follows:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Algebraic Tests for Symmetry Example. The graph of y = x3 – x is symmetric with respect to the origin because:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Circles A circle with center at (h, k) and radius r consists of all points (x, y) whose distance from (h, k) is r. From the Distance Formula, we have the standard equation of a circle as:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Circles Example. Find the standard form of the equation of the circle with center at (2, -5) and radius 4. (x-2) 2 +(y-(-5)) 2 =4 2 or (x-2) 2 +(y+5) 2 =16