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Chapter 2 Graphs
2.1 The distance and midpoint formulas
- Rectangular or Cartesian Coordinate System To locate a point on the real number line, we need a real number. For work in a two- dimensional plane, we locate points by using two numbers, an ordered pair, ( x, y ). Example 1 Locate the following points whose coordinates are: (1) (-3, 1) ( 2) (-2, -3) (3) (3, 2) The origin has coordinate (^0 ,^0 ). Any point on the x -axis has coordinates of the form ( x , 0 ), and any point on the y -axis has coordinates of the form ( 0 , y ).
- Distance Between Points Distance Formula The distance between two points P 1^ (^ x 1 , y 1 )and P 2^ (^ x 2 , y 2 ), defined by d^ (^ P 1 , P 2 ), is Example 2 Find the distance d between the points (^4 ,^5 )and (^3 ,^2 ).
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.
- Midpoint Formula The midpoint M^ (^ x , y )of the line segment from P 1 (^) ( x 1 , y 1 ) to P 2 (^) ( x 2 , y 2 )is Example 4 Find the midpoint of a line segment from P 1^ (^5 ,^5 )to P 2^ (^3 ,^1 ). Plot the points P 1^ and P 2^ , and their midpoint.
2.2 Graphs of Equations
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.
2.3 Lines
- Slope of a Line
- Equations of Lines A vertical line is given by an equation of the form , where a is the x - intercept. A horizontal line is given by an equation of the form , where b is the y -intercept. Point-Slope Form : An equation of a nonvertical line of slope m that contains the point (^ x 1^ , y 1 )is. Slope-Intercept Form : An equation of a line with slope m and y -intercept b is General Form : Example 4 Graph the equation x 3. Example 5 Find an equation of the line with slope 4 and containing the point (^1 ,^2 ). Example 6 Find an equation of the horizontal line containing the point (^3 ,^2 ).
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.
- Parallel Lines Two nonvertical lines are parallel if and only if their slopes are equal. Example 9 Show that the lines given by the following equations are parallel: 2x + 3y = 4 and 4x + 6y = 5
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.
