Try our Free Online Math Solver!

Rectangular Coordinate System,Graphs of Equations and Lines
Rectangular Coordinate System , Graphs of Equations and Lines
Section 3.1: Rectangular Coordinate System
· Note 1: The rectangular coordinate system is divide into 4 quadrants: I, II, III and IV.
· Note 2: Distance between two points A(x_{1},y_{1}) and B(x_{2},y_{2}) is:
Example 1: Find the distance between A(1, 2) and B(1, 4)
· Note 3: Midpoint of a line connecting 2 points A(x_{1},y_{1}) and B(x_{2},y_{2}):
Example 2: Find the midpoint of from A(1, 4) and B(1, 2)
Example 3: Find a formula that expresses the fact that an arbitrary point P(x,
y) is on the perpendicular bisector
l of segment AB: A(3 , 2) , B(5, 4)
More examples:
Example 4: Find all points on the yaxis that are a distance of 4 from P(3,5)
Example 5: Find all points on the xaxis that are a distance of 5 from P(2,3)
Section 3.2: Graphs of Equations
1) graph y = x^{2} + 5
2) graph
· Symmetric with respect to the yaxis, then f(x) = f(x)
such as: y = x^{2} + 2
· Symmetric with respect to the xaxis, then f(y) = f(y) such as: x = y^{2} + 3.
· Ssymmetric with respect to the origin, then simultaneous substitution of x
for x and y for y gets the same
equation such as: 2y = x^{3}
Examples: determine which graphs are symmetric with
respect of:
a) the yaxis b) the xaxis c) the origin
Equation Of The Circle With a Radius r :
8) Find the equation of the circle with center (4, 1) and
r = 3
9) Find the equation of the circle with center (2, 4) and passes (2 , 1)
10) Find the equation of the circle with center (4, 1) and tangent to the
xaxis.
11) Find the equation of the circle with end points of a diameter A(5, 2) and
B(3,6)
12) Find the center and the radius of x^{2} + y^{2} + 8x  10y + 37 = 0
13) Find the center and the radius of 9x^{2} + 9y^{2} + 12x  6y + 4 = 0
14) Find the center and the radius of x^{2} + y^{2} + 4x + 6y + 16 = 0
15) Graph the circle or semicircle: x^{2} + y^{2} = 8
16) Graph the circle or semicircle: (x  4)^{2} + (y + 2)^{2} = 4
17) Graph the circle or semicircle: x^{2} + y^{2} =16
18) Graph the circle or semicircle: (half the
circle of example 17 when y is isolated)
19) Graph the circle or semicircle: (half the circle of example 17 when x is isolated)
Important:
· If , then the graph
is the upper half of a circle (see example 18).
· If , then the graph is the lower half of a
circle.
· If , then the graph is the left half of a
circle (see example 19)
· If , then the graph is the right half of a
circle
Book, Exer. 57  60: Find the equations for the upper
half, lower half, right half, and left half of the
circle:
Book, Exer. 61  62: Determine whether the point P is
inside, outside, or on the circle with center C
and radius r :
62) (a) P(3,8), C(2,4), r = 13
(b) P(2,5), C(3,7), r = 6
(c) P(2,5), C(3,7), r = 6
Book, Exer. 63  64: For the given circle, find (a) the
x intercepts and (b) the yintercepts:
64) x^{2} + y^{2} 10x + 4y + 13 = 0
Section 3.3: Lines
1) Find the slope of the line passing the points (2, 4)
and (3, 1)
2) Find the slope of the line passing the points (2,3) and (1, 3)
· m > 0 or positive slope , then the line is increasing or
rising
m < 0 or negative slope , then the line is decreasing or falling
m = 0, then the line is horizontal
m = undefined, no slope, then the line is vertical
3) graph the line that passes the point P(2 , 4) and has m = 1/3
· Slope Intercept Equation:
y = mx + b (m is the slope, b is the yintercept)
Find the slope and the yintercept for:
4) 4x + 2y = 5
5) 3y  2x = 5 + 9y – 2x
6) 5x = 2/3 y  10
7) y = 10
8) x = 5
· General Linear Equation :
ax + by = c
Note: isolate y in the above equation and you will get:
Compare it to the SlopeIntercept Equation: slope = a/b and the yintercept = c/b.
Graphing a Linear Equation:
· Using the x and yintercepts ( xintercepts when y = 0 ; yintercepts when x =
0)
9) 4x  5y = 20
10) 3x + 2y = 12
11) 2y  3x = 0
· Using Slope Intercept Equation (isolate y to get y = mx + b)
12) 2y  3x = 0
13) 2x + y = 3
· Special Cases:
14) y = 3 15) x = 5 16) y = 0
Finding the equation of a line: y = mx + b (or ax + by = c)
· Case 1: One point is given and the slope:
Find the equation of the line having the given slope and containing the given
point:
17) m = 2 ; (2, 8) 18) m = 4/5 ; (2, 3)
· Case 2: Two points are given:
Find the equation of the line containing the given pair of points:
19) (2, 1) and (1, 3)
20) (1, 5) and (2 , 1)
Parallel and perpendicular lines:
· Two lines are parallel if they have same slope but
different y intercepts.
Example: y = 3x – 4 and y= 3x –2 where the slope is = 3 in both
· Two line are perpendicular if the slopes are m and –1/m
( multiplication of both slopes = 1)
Example: y = 2x  5 and y= 1/2 x + 3 where m = 2 and m = 1/2
Find an equation of the line containing the given point
and parallel to the given line:
21) (2, 1) ; 2y + 10 = x 22) (8, 4) ; 2y  2x = 17
Find an equation of the line containing the given point and perpendicular to the
given line:
23) (3, 4) ; y  3x = 2 24) (2, 3) ; 2y + 4x = 1
Find an equation of the perpendicular bisector of the segment AB:
25) A(4, 2) , B(2, 10)
Applications:
26) The cost of producing 100 units is $60 and for 120 units is $70. Find the
equation, and the cost
of 200 units
Prev  Next 