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Section P.3 9 The Cartesian Plane and Graphs of Equations Course Number Section P.3 The Cartesian Plane and Graphs of Equations Instructor Objective: In this lesson you learned how to plot points in the coordinate plane, use the Distance and Midpoint Formulas, and sketch graphs of equations. Important Vocabulary Date Define each term or concept. Rectangular coordinate system A plane (Cartesian plane) used to graphically represent ordered pairs of real numbers. Ordered pair Two real numbers x and y, written (x, y), which represent a point in the Cartesian plane. Graph of an equation The set of all points that are solutions of the equation. Intercepts The points at which a graph intersects the x- or y-axis. Symmetry If a graph is folded along a dividing line and the portion of the graph on one side of the dividing line coincides with the portion of the graph on the other side of the dividing line, then the graph is said to have symmetry. I. The Cartesian Plane (Pages 25−26) On the Cartesian plane, the horizontal real number line is usually called the , and the vertical real number line is x-axis usually called the y-axis of intersection . The origin is the point of these two axes, and the two axes divide the plane into four parts called quadrants . On the Cartesian plane shown below, label the x-axis, the y-axis, the origin, Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. y-axis 5 Quadrant II 1 -5 -3 Quadrant III Quadrant I 3 -1 -1 -3 origin 1 3 5 x-axis Quadrant IV -5 Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to plot points in the Cartesian plane 10 Chapter P To sketch a scatter plot of paired data given in a table, . . . 5 Prerequisites y represent each pair of values by an ordered pair and plot the 3 resulting points. 1 Example 1: Explain how to plot the ordered pair (3, − 2), and then plot it on the Cartesian plane provided. Imagine a vertical line through 3 on the x-axis and a horizontal line through − 2 on the y-axis. The intersection of these two lines is the point (3, − 2). II. The Distance Formula (Page 27) The Distance Formula states that . . . the distance d between the points (x1, y1) and (x2, y2) in the plane is d = √ (x2 − x1)2 + (y2 − y1)2 . -5 -3 -1 -1 1 3 5 x (3, -2) -3 -5 What you should learn How to use the Distance Formula to find the distance between two points Example 2: Explain how to use the Distance Formula to find the distance between the points (4, 2) and (5, − 1). Then find the distance and round to the nearest hundredth. Explanations will vary. Let x1 = 4 and y1 = 2, and let x2 = 5 and y2 = − 1. Then substitute these values into the Distance Formula and simplify; 3.16 III. The Midpoint Formula (Page 28) To find the midpoint of a line segment, find the values average of the respective coordinates of the two endpoints of the line segment using the Midpoint Formula. What you should learn How to use the Midpoint Formula to find the midpoint of a line segment The Midpoint Formula gives the midpoint of the segment joining the points (x1, y1) and (x2, y2) as . . . Midpoint = x1 + x2 y1 + y2 , 2 2 Example 3: Explain how to find the midpoint of the line segment with endpoints at (− 8, 2) and (6, − 10). Then find the coordinates of the midpoint. Explanations will vary. Find the average of the two x-coordinates and find the average of the two y-coordinates. These averages form the coordinates of the midpoint, (− 1, − 4). Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. Section P.3 11 The Cartesian Plane and Graphs of Equations IV. The Graph of an Equation (Page 29) What you should learn How to sketch graphs of equations To sketch the graph of an equation in two variables, . . . construct a table of values that consists of several solution points of the equation. Plot the solution points on a rectangular 5 coordinate system and connect the points with a smooth curve. y 3 Example 4: Complete the table. Then use the resulting solution points to sketch the graph of the equation y = 3 − 0.5x. x y −4 5 −2 4 0 3 2 2 4 1 1 -5 -3 -1 -1 1 3 -3 -5 V. Intercepts of a Graph (Page 30) An x-intercept is written as the ordered pair a y-intercept is written as the ordered pair To find x-intercepts, . . . , and (x, 0) (0, y) . What you should learn How to find x- and y-intercepts of graphs of equations let y be zero and solve the equation for x. To find y-intercepts, . . . let x be zero and solve the equation for y. Example 5: For the equation 3 x − 4 y = 12 , find: (a) the x-intercept(s), and (b) the y-intercept(s). (a) (4, 0) (b) (0, − 3) VI. Symmetry (Pages 31−33) The three types of symmetry that a graph can exhibit are . . . y-axis symmetry, origin symmetry, or x-axis symmetry. Knowing the symmetry of a graph before attempting to sketch it is helpful because . . . then you need only half as many solution points to sketch the graph. 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. 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. A graph is Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to use symmetry to sketch graphs of equations 5 x 12 Chapter P Prerequisites symmetric with respect to the origin if, whenever (x, y) is on the (− x, − y) graph, is also on the graph. 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. 35 30 The graph of an equation is symmetric with respect to the origin if . . . y 25 replacing x with − x and y with − y yields an equivalent 20 equation. 15 10 5 Example 6: Use intercepts and symmetry to sketch the graph of the equation y = 2 x 2 + 2 . 0 -5 VII. Circles (Page 33) (x − h)2 + (y − k)2 = r2 -1 -5 1 3 5 x What you should learn How to find equations and sketch graphs of circles The standard form of the equation of a circle with center (h, k) and radius r is -3 . The standard form of the equation of a circle with radius r and its center at the origin is x2 + y2 = r 2 . 5 Example 7: For the equation ( x + 2) + ( y − 1) = 4 , find the center and radius of the circle and then sketch the graph of the equation. 2 2 y 3 1 Center: (− 2, 1) Radius: 2 -5 -3 -1 -1 1 3 5 x -3 -5 Homework Assignment Page(s) Exercises Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved.