e - book for college algebra king fahd university of petroleum & minerals 2.1 e - book for...

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

2.1


Rectangular Coordinates

Distance Formula Graphs of Equations and Symmetry The Intercepts of a Graph Symmetry of a Graph Equation of a Circle Midpoint Formula



Introduction… cont…

a. The horizontal and vertical axes are often labeled the x-axis and y-axis, but other variables may sometimes be used to label the axis.

b. Usually the same unit of length is used on both axes, however, this need not to be always the case.

c. The coordinates a and b of a point P(a,b) not on one of the coordinate axis, satisfy the signs indicated in the following table:

Quadrant I Quadrant II Quadrant III Quadrant IV



Example 1Calculate the distance between and .

Distance Formula

The distance between two points and on the xy-coordinate plane is given by

2 2

2 1 2 1, d Q P d x x y y

2 25 2 3 6 9 81 90 d

¿3 √10



Example 2Determine whether the points , and are vertices of a right angled triangle.

2 2, 4 2 1 1 40 d D E

2 2, 4 3 1 4 10 d E F

2 2, 3 2 4 1 50 d D F

Since, , it follows from the converse of Pythagorean theorem that the triangle with vertices D, E and F is a right angled triangle, and that its right angle occurs at the vertex E.



Example 3Determine if and are points on the graph of

Because is not on the graph of

Because , is not on the graph of

Because , is on the graph of



Example 4 Graph the equation

We locate some points on the graph of the equation in the following table

Assign x

Calculate y Points on the graph



Example 5 Graph the equation

We locate some points on the graph of the equation in the following table

Points on the graph



The Intercepts of a Graph

The points (if any) at which the graph of an equation touches or crosses the y-axis are called the y-intercepts of the graph.

Similarly we define the x-intercepts of the graph. To find the x-intercepts if any, we let in the equation and solve for x and to find the y-intercepts, let in the equation and solve for y.



Example 6Find the x-intercepts and y-intercepts of the equations

To find the x-intercept(s), set

one x-intercept

For the y-intercept(s), set or

There are two y-intercepts and



Example 7 Find the x-intercepts and y-intercepts of the equations

For the x-intercept(s), set

There are two x-intercepts and



Example 7 Find the x-intercepts and y-intercepts of the equations

For the y-intercept(s), set

There are two y-intercepts and



Symmetry of a Graph

A graph is said to be symmetric

a. about x-axis if for every on the graph, the point is also on the graph.

b. about y-axis if for every point on the graph, the point is also on the graph.

c. about the origin if for every point on the graph, the point is also on the graph.



Example 8 Test the equations for symmetry

Replacing (x) by (–x) we get which is not an equivalent equation.Replacing by we get which is not an equivalent equation.Replace by to get which is an equivalent equation to the original equation.Thus the graph of is symmetric with respect to the origin.



Equation of Circle & Midpoint Formula

The equation of a circle is given by

The midpoint of the line segment joining the points and is given by



Example 9Write the equation of the circle with center and radius Also, find the midpoint of the line segment joining

The equation of the circle with center and radius is given by

While the midpoint of the line segment joining is



Example 10Write the equation

in the standard form, and then determine the radius and the center of the given circle.

By completing the square in both x and y separately, we get.

Therefore, the center is and the radius is

e - book for college algebra king fahd university of petroleum & minerals 2.1 e - book for...

Documents