E - BOOK FOR COLLEGE ALGEBRA
2.1
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Rectangular Coordinates
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
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Distance Formula
Graphs of Equations and Symmetry
The Intercepts of a Graph
Symmetry of a Graph
Equation of a Circle
Midpoint Formula
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E - BOOK FOR COLLEGE ALGEBRA
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
𝒂>𝟎
𝒃>𝟎
𝑎<0
𝑏>0
𝑎<0
𝑏<0
𝑎>0
𝑏<0
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E - BOOK FOR COLLEGE ALGEBRA
Distance Formula
The distance between two points 𝑄 𝑥1 , 𝑦1
𝑃 𝑥2 , 𝑦2 on the xy-coordinate plane is given by
d  Q, P   d 
Example 1
d 
 x2  x1 
2
  y2  y1 
and
2
Calculate the distance between 2, −6 and
5, 3 .
5  2
2
 3  6
2

9  81 
= 3 10
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90
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E - BOOK FOR COLLEGE ALGEBRA
Example 2
Determine whether the points 𝐷 −2, −1 ,
𝐸 4, 1 and 𝐹 3, 4 are vertices of a right
angled triangle.
d  D, E  
 4  2
2

40
d  E, F  
 4  3  1  4 
2
2
3

2

4

1

 


10

50
d  D, F  
2
 1  1
2
2
Since, ⅆ 𝐷, 𝐸 2 = ⅆ 𝐸, 𝐹 2 + ⅆ 𝐷, 𝐹 2 , 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.
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E - BOOK FOR COLLEGE ALGEBRA
Example 3
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Determine if −2, 3 , −1, 3 and 2, 5
are points on the graph of 𝑦 = 𝑥 2 + 1
Because 3 ≠ −2 2 + 1,
⇒ −2, 3 is not on the graph of 𝑦 = 𝑥 2 + 1
Because 3 ≠ −1 2 + 1,
⇒ −1, 3 is not on the graph of 𝑦 = 𝑥 2 + 1
Because 5 = 2 2 + 1,
⇒ 2, 5 is on the graph of
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𝑦 = 𝑥2 + 1
E - BOOK FOR COLLEGE ALGEBRA
Example 4
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Graph the equation 𝑦 = 2𝑥 + 5
We locate some points on the graph of the equation in
the following table
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 5
Graph the equation 𝑦 2 − 𝑥 = −1
We locate some points on the graph of the equation in
the following table
Points on
the graph
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E - BOOK FOR COLLEGE ALGEBRA
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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 𝑦 = 0 in the
equation and solve for x and to find the y-intercepts,
let 𝑥 = 0 in the equation and solve for y.
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E - BOOK FOR COLLEGE ALGEBRA
Example 6
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Find the x-intercepts and y-intercepts of the
equations 𝑦 2 = 𝑥 + 4
To find the x-intercept(s), set 𝑦 = 0
𝑥+4=0
𝑥 = −4
⟹ one x-intercept −4, 0
For the y-intercept(s), set 𝑥 = 0
𝑦2 − 4 = 0
⟹ 𝑦 = 2 or 𝑦 = −2
There are two y-intercepts 0, 2 and 0, −2
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 7
Find the x-intercepts and y-intercepts of the
equations 2𝑥 2 − 3𝑥 + 𝑦 2 − 𝑦 = 6
For the x-intercept(s), set 𝑦 = 0
2𝑥 2 − 3𝑥 = 6
2𝑥 2 − 3𝑥 − 6 = 0
𝑥=
3± 9+48
4
3± 57
𝑥=
4
⇒
There are two x-intercepts
3+ 57
,0
4
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and
3− 57
,0
4
E - BOOK FOR COLLEGE ALGEBRA
Example 7
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Find the x-intercepts and y-intercepts of the
equations 2𝑥 2 − 3𝑥 + 𝑦 2 − 𝑦 = 6
For the y-intercept(s), set 𝑥 = 0
𝑦2 − 𝑦 = 6
𝑦2 − 𝑦 − 6 = 0
𝑦+2 𝑦−3 =0
𝑦 = −2 or
𝑦=3
There are two y-intercepts
0, −2 and
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0, 3
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Symmetry of a Graph
A graph is said to be symmetric
a. about x-axis if for every 𝑥, 𝑦 on the graph, the
point 𝑥, −𝑦 is also on the graph.
b. about y-axis if for every point 𝑥, 𝑦 on the graph,
the point −𝑥, 𝑦 is also on the graph.
c. about the origin if for every point 𝑥, 𝑦 on the
graph, the point −𝑥, −𝑦 is also on the graph.
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 8
Test the equations for symmetry 𝑦 =
−𝑥 2 +4
−𝑥
Replacing (x) by (–x) we get 𝑦 =
which is not an equivalent equation.
Replacing (𝑦) by (– 𝑦) we get −𝑦 =
which is not an equivalent equation.
𝑥 2 +4
𝑥
⇒ 𝑦=
𝑥 2 +4
−𝑥
𝑥 2 +4
𝑥
−𝑥 2 +4
−𝑥
Replace (𝑥, 𝑦) by (– 𝑥, – 𝑦) to get −𝑦 =
which is an equivalent equation to the original equation.
Thus the graph of 𝑦 =
the origin.
𝑥 2 +4
𝑥
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is symmetric with respect to
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E - BOOK FOR COLLEGE ALGEBRA
Equation of Circle & Midpoint Formula
The equation of a circle is given by
𝑟2 = 𝑥 − ℎ
2
+ 𝑦−𝑘
2
The midpoint of the line segment joining the points
𝑄 𝑥1 , 𝑦1 and 𝑃 𝑥2 , 𝑦2 is given by
𝑥1 + 𝑥2 𝑦1 + 𝑦2
,
2
2
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E - BOOK FOR COLLEGE ALGEBRA
Example 9
Write the equation of the circle with center
−2, 5 and radius 7. Also, find the
midpoint of the line segment joining
2, −15 and 4, 5
The equation of the circle with center −2, 5 and
radius 7 is given by
𝑥 + 2 2 + 𝑦 − 3 2 = 49
While the midpoint of the line segment joining
2, −15 and 4, 5 is
2+4 −13+5
,
= 3, −5
2
2
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E - BOOK FOR COLLEGE ALGEBRA
Example 10
Write the equation
𝑥 2 + 𝑦 2 − 6𝑥 + 4𝑦 + 5 = 0
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
𝑥 2 − 6𝑥 + 9 + 𝑦 2 + 4𝑦 + 4 = 8
𝑥 − 3 2 + 𝑦 + 2 2 = 8.
Therefore, the center is 3, −2 and
the radius is 𝑟 = 2 2.
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