Download presentation

Presentation is loading. Please wait.

Published byHenry Kittle Modified over 2 years ago

1
Digital Lesson on Graphs of Equations

2
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

3
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

4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Sketch the graph of y = –2x + 3. 1. 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)

5
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Sketch the graph of y = –2x + 3. 2. Plot the points in the coordinate plane. 48 4 8 4 –4 x y xy(x, y) –27(–2, 7) –15(–1, 5) 03(0, 3) 11(1, 1) 2–1(2, –1) Example continued

6
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Sketch the graph of y = –2x + 3. 3. Connect the points with a straight line. 48 4 8 4 –4 x y Example continued

7
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 24 2 6 8 –2 Example: Sketch Graph (Quadratic Function)

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

9
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 –22 2 4 Example: Sketch Graph (Absolute Value Function)

10
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

11
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

12
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 2 + 4(0) – 5 = –5 So, the y-intercept is (0, –5). Example: Find Intercepts

13
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

14
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

15
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

16
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.

17
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

18
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:

19
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:

20
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:

21
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

Similar presentations

OK

2.2 Graphs of Equations in Two Variables Chapter 2 Section 2: Graphs of Equations in Two Variables In this section, we will… Determine if a given ordered.

2.2 Graphs of Equations in Two Variables Chapter 2 Section 2: Graphs of Equations in Two Variables In this section, we will… Determine if a given ordered.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on product advertising pitch Ppt on mahatma gandhi life in hindi Ppt on area of parallelogram and triangles for class 9 Ppt on energy resources and conservation Ppt on kingdom monera examples Ppt on abortion Ppt on basic leadership skills Ppt on mauryan administration Ppt on entrepreneurship development Ppt on human resources planning