Section P.3 the Cartesian Plane and Graphs of Equations 9

Section P.3 The Cartesian Plane and Graphs of Equations 9

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, Date and sketch graphs of equations.

Important Vocabulary 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) What you should learn How to plot points in the On the Cartesian plane, the horizontal real number line is usually Cartesian plane called the x-axis , and the vertical real number line is usually called the y-axis . The origin is the point of intersection 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

Quadrant II 3 Quadrant I

1 origin x-axis -5 -3 -1 1 3 5 -1

Quadrant III -3 Quadrant IV

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To sketch a scatter plot of paired data given in a table, . . . y 5 represent each pair of values by an ordered pair and plot the 3 resulting points. 1 x -5 -3 -1 1 3 5 Example 1: Explain how to plot the ordered pair (3, - 2), and -1 then plot it on the Cartesian plane provided. (3, -2) Imagine a vertical line through 3 on the x-axis and -3 a horizontal line through - 2 on the y-axis. The intersection of these two lines is the point (3, - 2). -5

II. The Distance Formula (Page 27) What you should learn The Distance Formula states that . . . the distance d between How to use the Distance Formula to find the the points (x1, y1) and (x2, y2) in the plane is distance between two 2 2 points d = Ö (x2 - x1) + (y2 - y1) .

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) What you should learn How to use the Midpoint To find the midpoint of a line segment, find the average Formula to find the values of the respective coordinates of the two midpoint of a line segment endpoints of the line segment using the Midpoint Formula. The Midpoint Formula gives the midpoint of the segment joining the points (x1, y1) and (x2, y2) as . . .

æ x1 + x2 y1 + y2 ö Midpoint = ½ ¾¾¾¾ , ¾¾¾¾ ÷ è 2 2 ø

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).

IV. The Graph of an Equation (Page 29) What you should learn How to sketch graphs of To sketch the graph of an equation in two variables, . . . equations construct a table of values that consists of several solution points of the equation. Plot the solution points on a rectangular y coordinate system and connect the points with a smooth curve. 5

3 Example 4: Complete the table. Then use the resulting solution points to sketch the graph of the equation 1 y = 3 - 0.5x. x -5 -3 -1 1 3 5 -1 x - 4 - 2 0 2 4 -3 y 5 4 3 2 1

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V. Intercepts of a Graph (Page 30) What you should learn How to find x- and An x-intercept is written as the ordered pair (x, 0) , and y-intercepts of graphs of a y-intercept is written as the ordered pair (0, y) . equations To find x-intercepts, . . . 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 3x - 4y =12 , find: (a) the x-intercept(s), and (b) the y-intercept(s). (a) (4, 0) (b) (0, - 3)

VI. Symmetry (Pages 31-33) What you should learn How to use symmetry to The three types of symmetry that a graph can exhibit are . . . sketch graphs of y-axis symmetry, origin symmetry, or x-axis symmetry. equations

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

symmetric with respect to the origin if, whenever (x, y) is on the graph, (- x, - y) 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. y 35 The graph of an equation is symmetric with respect to the origin 30 25 if . . . replacing x with - x and y with - y yields an equivalent 20 equation. 15 10 Example 6: Use intercepts and symmetry to sketch the graph 5 2 0 x of the equation y = 2x + 2 . -5 -3 -1 1 3 5 -5

VII. Circles (Page 33) What you should learn How to find equations The standard form of the equation of a circle with center and sketch graphs of (h, k) and radius r is (x - h)2 + (y - k)2 = r2 . circles

The standard form of the equation of a circle with radius r and its center at the origin is x2 + y2 = r2 .

y 5 Example 7: For the equation (x + 2)2 + ( y -1) 2 = 4 , find the

center and radius of the circle and then sketch the 3 graph of the equation. 1 x Center: (- 2, 1) -5 -3 -1 1 3 5 Radius: 2 -1

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