e - book for college algebra king fahd university of petroleum & minerals 2.1 e - book for...
TRANSCRIPT
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 20013 All Right Reserved
2.1
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Rectangular Coordinates
Distance Formula Graphs of Equations and Symmetry The Intercepts of a Graph Symmetry of a Graph Equation of a Circle Midpoint Formula
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 5 Graph the equation
We locate some points on the graph of the equation in the following table
Points on the graph
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 7 Find the x-intercepts and y-intercepts of the equations
For the x-intercept(s), set
There are two x-intercepts and
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 7 Find the x-intercepts and y-intercepts of the equations
For the y-intercept(s), set
There are two y-intercepts and
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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