The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.

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Laureen Mosley
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1 We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section 1.1 Cartesian Plane: The Cartesian plane (or simply the xy-plane) is shown below. Familiarize yourself with its parts. Remember a point s coordinates are alphabetical, x then y or (x, y). The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is. Graphs of Equations: The graph of an equation in two variables shows us every point (x, y) that satisfies the equation (or makes it true). Consider the equation y = x 3. Complete the table below and then look at the graph I have provided. x y = x What does the equation say about x and y? How are they related? Every point shows this relationship in their points (x, y). 1 3 Every y value is gotten by the same operations. A complete graph shows all the intercepts and turns. 1

2 expl 1: Determine if the ordered pairs are solutions to thee following equation. x 3y 15 (3, ), (0, 5), (4, 1) What does it mean for an ordered pair to be a solution? Distance Formula: Below are two points on a plane. If we want to find the distance between them, we can draw in a right triangle (picture on right) and use the Pythagorean Theorem. Notice in particular how the legs of the triangle are figured. The vertical leg turns out to be 8 1 or the difference of the y-values. Why is the horizontal leg the difference of the x-values? Could you use this triangle to find the distance between the two points?

3 We will now use two points in general to derive the formula for the distance between them. The procedure is similar to the example above. first point (x 1, y 1 ) second point (x, y ) Notice how the legs are figured. Pythagorean Theorem: c = a + b What are a, b, and c? Let s derive the distance formula. We start with the Pythagorean Theorem and replace the variables with the pieces of our triangle. c = a + b Put d in for the hypotenuse, and the legs in for a and b. Then square root both sides to get d alone. 3

4 expl : Use the distance formula to find the distance between the two points (5, 4) and (10, 6). Give an exact distance and a three-decimal place approximation. d x x y 1 y1 Your points are (x 1, y 1 ) and (x, y ). Mind the order of operations! Reduce the radical to get the exact distance. Use your calculator for the approximation. expl 3: The points (-3, -1) and (9, 4) are the endpoints of the diameter of a circle. Find the length of the radius of the circle. Round to three decimal places if needed. 4

5 Midpoint Formula: This gives the point in the exact middle of two other points. (It is said to be the midpoint of the segment connecting the two points.) See the picture below. If we define the points as (x 1, y 1 ) and (x, y ), can you come up with the midpoint formula? Midpoint lies halfway between x-values and halfway between y-values. Midpoint is the average of the two points. What could the formula be? expl 4: Find the midpoint of the line segment whose endpoints are (9, -5) and (4, 3). Also quickly plot the two points with your midpoint to check yourself. 5

6 Finding x- and y-intercepts: The x- and y-intercepts of a graph are where the graph intersects the x and y axes. x and y intercepts occur on the axes. The one thing you know about any point on an axis, is that the other coordinate is zero. We ll use that with the equation to find the x and y intercepts. expl 5: Find the intercepts of y x 4 = 0. Graph on your calculator to check. y-intercept: Set x to 0 and solve for y. x-intercept: Set y to 0 and solve for x. To graph, solve the equation for y. Don t forget the. 6

7 Equations of Circles: A circle is the set of all points (x, y) equidistant from a single point (h, k) called the center. Using r for the radius (the distance between the center and any point on the circle) and the distance formula, we get the equation r x h y k and you get the following. x h The equation of a circle is given as is the radius. y k r. Square both sides. The point (h, k) is the center and r expl 6: Find the center and radius of the circle below. Then graph it on the calculator. x 6 y 3 5 When you graph it, select Zoom Standard first, then switch it to Zoom Square. This will make it more circular looking. expl 7: Find the equation of a circle that has a center of (6, -5) and passes through the point (1, 7). The point (1, 7) is on the circle. The equation of a circle is. What do we need to know? 7

8 Worksheet: Things to know about your calculator: This calculator is a laundry list of things I have useful over the years. Try out everything it mentions. Instructions for the TI83 should work for the TI84 calculators. Worksheet: Getting started on your graphing calculator: This is an introduction to home screen operations, graphing, window settings, and the TRACE function. Instructions for the TI83 should work for the TI84 calculators. 8

Each point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.

Each point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P. Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations