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by: Marquise Graham


Marquise Graham
GPA 3.8

Hung Ngoc Nguyen

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Hung Ngoc Nguyen
Class Notes
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This 15 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Hung Ngoc Nguyen in Fall. Since its upload, it has received 19 views. For similar materials see /class/207047/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.


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Date Created: 09/19/15
Lecture 7 Part I Section 11 Rectangular Coordinates Def The retangular or Cartesian coordinate system is formed by coordinate axes horizontal L aXlS and vertical y axis The origin is the point of intersection of coordinate axes the axes divide the coordinate plane or my plane into four quadrants Ill W IV Each point P in the cry plane corresponds to an ordered pair 51 y of real numbers called the coordinates of P T0 plot P 51 y IJ coordihate directed distance of the point from the y axis y coordihate directed distance of the point from the L aXlS Plot P1 3 2 and P2 1 3 Locate all points which are 2 units above the x axis In which quadrahts if any do they lie Locate all points with x coordihate 0 Pythagorean Theorem For a right triangle with hypotenuse of length C and sides of lengths a and b we have a2 b2 02 NOTE The converse of the theorem is also true That is if a2 b2 02 then the triangle is a right triangle Distance Formula The distance between two points A 131 and B x2 32 is given by dA B g If P 14 and Q 4 3 nd the distance between P and Q Midpoints Def A point M on the line segment With end points A and B is the midpoint of the segment if dA M KM B Midpoint Formula The midpoint M of the line segment With endpoints A 5131791 and B 2ay21S may The point x 2 is the midpoint of the segment With endpoints 2 1 and 3 3 Find 1 and y Practice 1 Find all points having an x coordinate of 2 Whose distance from the point 5 1 is 5 2 Show that the points 10 11 and 2 7 are the vertices of a right triangle Lecture 7 Part II Section 12 Graph of Equations An equation in two variables a and y is a starnent in which two expressions involving 1 and y are equal An ordered pair a b is a solution of an equation in two variables if the equation is true when a is substituted for 1 and b is substituted for y The graph of an equation in two variables is the set of points in the cry plane which are solutions to the equation Graph by Plotting Points 1 If possible rewrite the equation so that one of the variable is isolated on one side of the equation 2 Make a table of values showing several solution points 3 Plot these points on a rectangular coordinate system 4 Connect the points with a smooth curve or line Graph by plotting points 1 2x y 3 Intercepts Def The points at which a graph intersects crosses or touches the coordinate axes are called intercepts An xintercept is the x coordinate of a point at which the graph intersects the L aXiS To nd it An yintercept is the y coordinate of a point at which the graph intersects the y axis To nd it Find the intercepts of the equation and graph 3 2x 3 Symmetry Def A graph is symmetric with respect to the xaxis if for every point 51y on the graph the point x y is also on the graph Algebraic test The graph of an equation is sym metric with respect to the x axis if replacing y with y in the equation yields an equivalent equation Def A graph is symmetric with respect to the yaxis if for every point 51y on the graph the point 1331 is also on the graph Algebraic test The graph of an equation is sym metric with respect to the y axis if replacing 1 with x in the equation yields an equivalent equation Def A graph is symmetric with respect to the origin if for every point x 3 on the graph the point 51 y is also on the graph Algebraic test The graph of an equation is sym metric with respect to the origin if replacing 1 with x and y with y in the equation yields an equivalent equation Test for symmetry and graph the following 1yll 1 2 1 25L y Circles Def A circle is a set of points in the cry plane that are a xed distance 7 called the radius from a xed point h k the center of the circle Find a relationship between 1 and y if P x y is any point on the circle Standard Form of the equation of a circle With radius 7 and center h 16 NOTE The unit circle is centered at the origin With radius 1 Its equation Describe the graph of x 32 y 42 25 Write the equation of the circle with center 1 2 and containing the origin Sketch the circle General form of the equation of a circle x2y2axbyC0 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 Find the center and radius of the circle 2x22y28x 4y90 Practice Find the equation of the Circle that has a diameter with endpoints 3 5 and 5 1


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