The Coordinate Plane<br />If two copies of the number line, one horizontal and one vertical, are placed so that they intersect at the zero point of each line, a pair of axes is formed.<br />The horizontal number line is called the x-axis and the vertical number line is called the y-axis.<br />The point where the lines intersect is called the origin.<br />We call this a rectangular coordinate plane or a Cartesian coordinate system.<br />
Inequalities in Two Dimensions<br />The graph of an inequality in two variables consists of all ordered pairs that make the inequality a true statement.<br />Example: Suppose we want to <br />graph the inequality .<br />Procedure:<br /><ul><li>Graph the boundary curve .
Draw a dashed curve if equality is not included.
Determine which region(s) formed by the curve makes the inequality true by testing with one point from inside each region.
Shade the region(s) that make the inequality true.</li></li></ul><li>Inequalities in Two Dimensions<br />Systems of two inequalities:<br />Idea: Graph both inequalities<br />and the region that has been <br />shaded in twice is the region we’re looking for.<br />Example at left: Graph the solution set of the following system of inequalities:<br />Click here to see a dynamicexample of linear inequalities.<br />
Distance<br />Suppose two points P1 and P2 have coordinates . What is the distance between P1 and P2?<br /><ul><li> The distance from P1 to P2 isthe length of the hypotenuseof a right triangle.
The length of the bottom sideis the same as the distance between x1 and x2 on the x-axis, </li></ul>that is, . The length of the vertical side is the same as the distance between y1 and y2, that is .<br />
Distance<br />So if we let d be the distance between P1 and P2, by the Pythagorean Theorem ….<br />Now we take the square rootof both sides. Since distance is positive:<br />Since we’re squaring in there,we can dispense with theabsolute values and get<br />Distance Formula<br />
Distance<br />Example: Find the distance between the points (4,-7) and (-1,3).<br />
Distance<br />See Mathematica Player demo on distance.<br />
Midpoint<br />The midpoint of the line segment connecting the points P1(x1,y1) and P2(x2,y2) is computed by simply averaging the x- and y-coordinates separately.<br />Midpoint FormulaThe midpoint between (x1,y1) and (x2,y2) is<br />Take a moment to find the coordinates of the point half way between and .<br />Answer: <br />
Circles in the Plane<br />A circle is defined as the set of all points that are the same distance from a given point.<br />The distance is called the radius.<br />The given point is called the center of the circle.<br />Let (x, y) be any point on a circle with center (h, k) and radius r as shown at left.<br />Since (x, y) must be r units from the center of the circle, the distance formula gives<br />
Standard Form of the Equation of a Circle<br />The graph of<br />is a circle of radius r (r ≥ 0) with center at the point (h, k). If the circle has center at the origin, the equation becomes<br />The circle is called the unit circle.<br />
Circles in the Plane<br />Example:<br />The center is at the point <br />(2, 3) and the radius is 1 unit.<br />We can figure out some points on the circle by starting with the center point, (2, 3), and adding or subtracting 1 from each coordinate.<br />So four examples would be <br />(3, 3), (1, 3), (2, 2), and (2, 4).Try: Find an equation of a circle with center (-3, 6) and radius 4.<br />Answer: <br />
Completing the square<br />Example: Find the center and radius of the circle whose equation is .<br />Since there is an x2-term and an x-term, we have to combine them to form the term by completing the square:<br />We first group the x-terms together:<br />To turn the thing in parentheses into a perfect square, divide the coefficient of the x-term by 2, square that number and add it to both sides.<br />
Completing the Square<br />We can then factor the x-terms:<br />Or to put it in the form of a circle:<br />We see that h is 1 and k is 0. So the center of the circle is (1,0) and the radius is 2.<br />
Completing the Square<br />To try on your own: Graph the equation<br />Hint: Divide first by 2.<br />Circle is centered at (-2,3) and hasradius 1.<br />