Chapter 2 Graphs
2.1 The distance and midpoint formulas
1. 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. Any point on the x-axis has coordinates of the form , and any point on the y-axis has coordinates of the form .
2. Distance Between Points
Distance Formula
The distance between two points and , defined by , is
Example 2 Find the distance d between the points and .
Example 3 Consider the three points , and .
- 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.
3. Midpoint Formula
The midpoint of the line segment from to is
Example 4 Find the midpoint of a line segment from to . Plot the points and , and their midpoint.
2.2 Graphs of Equations
Definition 1An equation in two variables, say x and y, is a statement in which two expressions involving x and y are equal. The expressions are called the sides of the equation. Since an equation is a statement, it may be true or false, depending on the value of the variables. Any values of x and y that result in a true statement are said to satisfy the equation.
Definition 2The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates satisfy the equation.
Example 1Determine if the following points are on the graph of the equation
(1) (2)
Example 2 Graph the equation:
Example 3 Graph the equation:
Example 4 Graph the equation:
Definition 3The 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 5Find 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 .
Definition 4
(1)A graph is said to be symmetric with respect to the x-axis if, for every point on the graph, the point is also on the graph.
(2)A graph is said to be symmetric with respect to the y-axis if, for every point on the graph, the point is also on the graph.
(3)A graph is said to be symmetric with respect to the origin if, for every point on the graph, the point 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) Find the intercepts. (2) Test for symmetry.
2.3 Lines
1. Slope of a Line
Let and be two distinct points. If , the slope m of the nonvertical line L containing P and Q is defined by
If , L is a vertical line and the slope m of L is undefined.
Remark .
Example 1 Find the slope m of the line containing the points and .
Example 2Compute the slopes of the lines and containing the following pairs of points.
Example 3 Draw a graph of the line that contains the points and has a slope of
(a) (b)
2. 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 is .
- Slope-Intercept Form: An equation of a line with slope m and y-intercept b is
- General Form:
Example 4Graph the equation.
Example 5Find an equation of the line with slope 4 and containing the point .
Example 6 Find an equation of the horizontal line containing the point .
Example 7 Find an equation of the line containing the points and .
Example 8 Find the slope m and y-intercept b of the line with equation .
3. 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
Example 10 Find an equation for the line that contains the points and is parallel to the line .
4. Perpendicular Lines
Two nonvertical lines are perpendicular if and only if the product of their slopes is -1.
Example 3 Find an equation of the line containing the point that is perpendicular to the line .
2.4 Circles
Definition 1 A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point . The fixed distance r is called the radius, and the fixed point is called the center of the circle.
The standard form of an equation of a circle with radius r and center is
The standard form of an equation of a circle of radius r with center at is
If the radius , 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 .
Example 2 Graph the equation.
Example 3 Find the center and radius of .
Definition 2 The equation is the general form of the equation of a circle.
Example 4 Graph the equation.