Section 1.3 -- The Coordinate Plane
- 1. MA107 PreCalculusSection 1.3The Coordinate Plane
- 2. The Coordinate PlaneIf 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.The horizontal number line is called the x-axis and the vertical number line is called the y-axis.The point where the lines intersect is called the origin.We call this a rectangular coordinate plane or a Cartesian coordinate system.
- 5. Inequalities in Two DimensionsThe graph of an inequality in two variables consists of all ordered pairs that make the inequality a true statement.Example: Suppose we want to graph the inequality .Procedure:Graph the boundary curve .
- 6. Draw a solid curve if equality is included.
- 7. Draw a dashed curve if equality is not included.
- 8. Determine which region(s) formed by the curve makes the inequality true by testing with one point from inside each region.
- 9. Shade the region(s) that make the inequality true.Inequalities in Two DimensionsSystems of two inequalities:Idea: Graph both inequalitiesand the region that has been shaded in twice is the region we’re looking for.Example at left: Graph the solution set of the following system of inequalities:Click here to see a dynamicexample of linear inequalities.
- 10. DistanceSuppose two points P1 and P2 have coordinates . What is the distance between P1 and P2? The distance from P1 to P2 isthe length of the hypotenuseof a right triangle.
- 11. The length of the bottom sideis the same as the distance between x1 and x2 on the x-axis, that is, . The length of the vertical side is the same as the distance between y1 and y2, that is .
- 12. DistanceSo if we let d be the distance between P1 and P2, by the Pythagorean Theorem ….Now we take the square rootof both sides. Since distance is positive:Since we’re squaring in there,we can dispense with theabsolute values and getDistance Formula
- 13. DistanceExample: Find the distance between the points (4,-7) and (-1,3).
- 14. DistanceSee Mathematica Player demo on distance.
- 15. MidpointThe 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.Midpoint FormulaThe midpoint between (x1,y1) and (x2,y2) isTake a moment to find the coordinates of the point half way between and .Answer:
- 16. Circles in the PlaneA circle is defined as the set of all points that are the same distance from a given point.The distance is called the radius.The given point is called the center of the circle.Let (x, y) be any point on a circle with center (h, k) and radius r as shown at left.Since (x, y) must be r units from the center of the circle, the distance formula gives
- 17. Standard Form of the Equation of a CircleThe graph ofis a circle of radius r (r ≥ 0) with center at the point (h, k). If the circle has center at the origin, the equation becomesThe circle is called the unit circle.
- 18. Circles in the PlaneExample:The center is at the point (2, 3) and the radius is 1 unit.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.So four examples would be (3, 3), (1, 3), (2, 2), and (2, 4).Try: Find an equation of a circle with center (-3, 6) and radius 4.Answer:
- 19. Completing the squareExample: Find the center and radius of the circle whose equation is .Since there is an x2-term and an x-term, we have to combine them to form the term by completing the square:We first group the x-terms together: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.
- 20. Completing the SquareWe can then factor the x-terms:Or to put it in the form of a circle:We see that h is 1 and k is 0. So the center of the circle is (1,0) and the radius is 2.
- 21. Completing the SquareTo try on your own: Graph the equationHint: Divide first by 2.Circle is centered at (-2,3) and hasradius 1.
- 22. Homeworkpg. 19: 1-19 odd, 25-43 odd, 51, 55, 57