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The basics of graphing in a two-dimensional plane, including the distance formula, midpoint formula, equations of lines and circles, and symmetry. Topics include locating points in a coordinate system, finding the distance between points, graphing equations, and identifying intercepts and symmetry properties.
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Example 3 Consider the three points A^ (^2 ,^1 ), B^ (^2 ,^3 )and C^ (^3 ,^1 ). Plot each point and form the triangle ABC. Find the length of each side of the triangle. Verify that the triangle is a right triangle. Find the area of the triangle.
Definition 3 The points, if any, at which a graph crosses or touches the coordinate axes are called the intercepts. The x -coordinate of a point at which the graph crosses or touches the x -axis is an x -intercept , and the y -coordinate of a point at which the graph crosses or touches the y- axis is a y -intercept. Example 5 Find the intercepts of the graph. What are its x -intercepts? What are its y - intercepts? Procedure for Finding Intercepts To find the x -intercept, let y=0 in the equation and solve for x. To find the y -intercept, let x=0 in the equation and solve for y. Example 6 Find the x -intercept(s) and the y -intercept(s) of the graph of y^ x^2 ^4.
Definition 4 (1) A graph is said to be symmetric with respect to the x -axis if, for every point ( x , y )on the graph, the point ( x , y ) is also on the graph. (2) A graph is said to be symmetric with respect to the y -axis if, for every point ( x , y )on the graph, the point ( x , y )is also on the graph. (3) A graph is said to be symmetric with respect to the origin if, for every point ( x , y )on the graph, the point ( x , y ) is also on the graph. To test the graph of an equation for symmetry with respect to the (1) x- axis: Replace y by – y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the x -axis. (2) y- axis: Replace x by – x in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the y -axis. (3) Origin: Replace x by – x and y by – y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin. Example 7 For the equation 1
2 2
x x y (1) Find the intercepts. (2) Test for symmetry.
Example 7 Find an equation of the line containing the points (^2 ,^3 ) and (^4 ,^5 ). Example 8 Find the slope m and y -intercept b of the line with equation 2 x^ ^4 y ^8.
The standard form of an equation of a circle with radius r and center (^ h ,^ k )is The standard form of an equation of a circle of radius r with center at (^0 ,^0 ) is If the radius (^) r 1 , the circle whose center is at the origin is called the unit circle and has the equation Example 1 Write the standard form of the equation of the circle with radius 5 and center ( 3 , 6 ). Example 2 Graph the equation (^3 ) (^2 )^16 x 2 y ^2 . Example 3 Find the center and radius of ( x 3 )^2 ( y 2 )^2 16. Definition 2 The equation x^^2 ^ y^2 ax by c ^0 is the general form of the equation of a circle. Example 4 Graph the equation x^^2 ^ y^2 ^4 x ^6 y ^12 ^0.