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Material Type: Notes; Professor: Nguyen; Class: PRECALC ALG & TRIG; Subject: Math: Calculus & Pre-calculus; University: University of Florida; Term: Spring 2009;
Typology: Study notes
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Lecture 7, Part I: Section 1. Rectangular Coordinates
Def. The retangular or Cartesian coordinate system is formed by coordinate axes: horizontal ๐ฅ-axis and vertical ๐ฆ-axis.
The origin is the point of intersection of coordinate axes. the axes divide the coordinate plane or ๐ฅ๐ฆ- plane into four quadrants.
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I
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Each point ๐ in the ๐ฅ๐ฆ-plane corresponds to an ordered pair (๐ฅ, ๐ฆ) of real numbers called the coordinates of ๐.
To plot ๐ = (๐ฅ, ๐ฆ):
๐ฅ-coordinate: directed distance of the point from the ๐ฆ-axis
๐ฆ-coordinate: directed distance of the point from the ๐ฅ-axis
ex. Plot ๐ 1 = (โ 3 , โ2) and ๐ 2 = (1, 3).
ex. Locate all points which are 2 units above the ๐ฅ-axis. In which quadrant(s), if any, do they lie?
ex. Locate all points with ๐ฅ-coordinate 0.
ex. If ๐ = (โ 2 , 4) and ๐ = (4, โ3), find the distance between ๐ and ๐.
Midpoints
Def. A point ๐ on the line segment with end- points ๐ด and ๐ต is the midpoint of the segment if ๐(๐ด, ๐ ) = ๐(๐, ๐ต).
Midpoint Formula
The midpoint ๐ of the line segment with endpoints ๐ด = (๐ฅ 1 , ๐ฆ 1 ) and ๐ต = (๐ฅ 2 , ๐ฆ 2 ) is
(๐ฅ, ๐ฆ) =
ex. The point (๐ฅ, โ2) is the midpoint of the segment with endpoints (2, โ1) and (3, ๐ฆ). Find ๐ฅ and ๐ฆ.
Practice.
Find all points having an ๐ฅ-coordinate of 2 whose distance from the point (5, โ1) is 5.
Show that the points (1, 0), (โ 1 , 1) and (2, 7) are the vertices of a right triangle.
ex. Graph by plotting points:
Intercepts
Def. The point(s) at which a graph intersects (crosses or touches) the coordinate axes are called intercepts.
An ๐ฅ-intercept is the ๐ฅ-coordinate of a point at which the graph intersects the ๐ฅ-axis. To find it,
An ๐ฆ-intercept is the ๐ฆ-coordinate of a point at which the graph intersects the ๐ฆ-axis. To find it,
ex. Find the intercepts of the equation and graph: ๐ฆ = 2๐ฅ + 3
Def. A graph is symmetric with respect to the origin if for every point (๐ฅ, ๐ฆ) on the graph, the point (โ๐ฅ, โ๐ฆ) is also on the graph.
Algebraic test: The graph of an equation is sym- metric with respect to the origin if replacing ๐ฅ with โ๐ฅ and ๐ฆ with โ๐ฆ in the equation yields an equivalent equation.
ex. Test for symmetry and graph the following.
ex. Describe the graph of (๐ฅ โ 3)^2 + (๐ฆ + 4)^2 = 25.
ex. Write the equation of the circle with center (1, โ2) and containing the origin. Sketch the circle.
General form of the equation of a circle:
๐ฅ^2 + ๐ฆ^2 + ๐๐ฅ + ๐๐ฆ + ๐ = 0
NOTE: If an equation of a circle is in the general form, we use the method of completing the square to put the equation in standard form so that we can identify its center and radius.
ex. Find the center and radius of the circle
2 ๐ฅ^2 + 2๐ฆ^2 + 8๐ฅ โ 4 ๐ฆ + 9 = 0.
