# Coordinate Planes and Graphs

A rectangular coordinate system is a pair of perpendicular coordinate lines, called coordinate axes, which are placed So that they intersect at their origins. The labeling of axes with letters x and y is a common convention, but any letters may be used. If the letter x and y are used to label the coordinate axes, then the resulting plane is called xy-plane. In applications it is common to use letters other than x and y is shown In the following figures, as uv-plane and ts-plane. #### Ordered pair

By an ordered pair of real numbers we mean two real numbers in an assigned order. Every point P in a coordinate plane can be associated with a unique ordered pair of real numbers by drawing two lines through P, one perpendicular to the x-axis and the other to the y-axis. For example if we take (a,b)=(4,3), then on coordinate plane To plot a point P(a,b) means to locate the point with coordinates (a,b) in a coordinate plane. For example, different points are plotted. In a rectangular coordinate system the coordinate axes divide the plane into four regions called quadrants. These are numbered counterclockwise with roman numerals as shown #### Definition of Graph

The graph of an equation in two variables x and y is the set of points in the xy- plane whose coordinates are members of the solution set of that equation

Example: Sketch the graph of y = x2

 x y = x2 (x,y) 0 0 (0,0) 1 1 (1,1) 2 4 (2,4) 3 9 (3,9) -1 1 (-1,1) -2 4 (-2,4) -3 9 (-3,9)

This is an approximation to the graph of y = x2. In general, it is only with techniques

from calculus that the true shape of a graph can be ascertained. Example: Sketch the graph of y = 1/x

 X y=1/x (x,y) 1/3 3 (1/3,3) 1/2 2 (1/2,2) 1 1 (1 ,1) 2 1/2 (2,1/2) 3 1/3 (3,1/3) -1/3 -3 (-1/3 , -3) -1/2 -2 (-1/2 , -2) -1 -1 (-1 , -1) -2 -1/2 (-2, -1/2) -3 -1/3 (-3,-1/3)

Because 1/x is undefined when x=0, we can plot only points for which x ≠0 Example: Find all intercepts of
(a) 3x + 2y = 6
(b) x = y2-2y
(c) y = 1/x

Solution:

3x + 2y = 6 x-intercepts

Set y = 0 and solve for x 3x = 6   or   x = 2

is the required x-intercept. is the required y-intercept.

Similarly you can solve part (b), the part (c) is solved here

y = 1/x

x-intercepts

Set y = 0

1/x = 0 => x is undefined. Then, no y-intercepts

Set x = 0

y = 1/0 => y is undefined => No y-intercept

In the following figure, the points (x,y), (-x,y),(x,-y) and (-x,-y) form the corners of a rectangle. • symmetric about the x-axis if for each point (x,y) on the graph the point (x,-y) is also on the graph.

• symmetric about the y-axis if for each point (x,y) on the graph the point (-x,y) is also on the graph.

• symmetric about the origin, if for each point (x,y) on the graph the point (-x,-y) is also on the graph.  Definition:

The graph in the xy-plane of a function f is defined to be the graph of the equation y = f(x)

Example: 1

Sketch the graph of f(x) = x + 2

y = x + 2 graph of f(x) = x + 2

Example: 2 Sketch the graph of f(x) = |x|

y = |x|

 |x| = x if x ≥ 0, i.e. x is non-negative -x if x < 0, i.e. x is negative

The graph coincides with the line y = x         for x> 0 and with line y = -x

for x < 0 .  graph of f(x) = -x

On combining these two graphs, we get graph of f(x) = |x|

Example: 4 Sketch the graph of

t(x) = (x2- 4)/(x - 2) =

= ((x - 2)(x + 2)/(x - 2)) =

= (x + 2)       x ≠ 2

Hence this function can be written as

y = x + 2            x ≠ 2 Graph of h(x)= x2 - 4 Or                     x - 2

graph of y = x + 2 x ≠ 2

Example: 4 Sketch the graph of

 g(x) = 1      if x ≤ 2 x + 2      if x > 2 Graphing functions by Translations

- Suppose the graph of f(x) is known

- Then we can find the graphs of

y = f(x) + c

y = f(x) - c

y = f(x + c)

y = f(x - c)

y = f(x) + c         graph of f(x) translates

UP by c units

y = f(x) - c          graph of f(x) translates

DOWN by c units

y = f(x + c)         graph of f(x) translates

LEFT by c units

y = f(x - c)          graph of f(x) translates

RIGHT by c units

Example: 5 Sketch the

graph of y = f(x) = |x - 3| + 2

Translate the graph of y = |x| 3 units to the RIGHT to get the graph of

y = |x-3|

Translate the graph of y = |x - 3| 2 units to the UP to get the graph of y = |x - 3| + 2 Example: 8

Sketch the graph of

y = x2 - 4x + 5

- complete the square

y + 4 = (x2 - 4x + 5) + 4 y = (x2 - 4x + 4) + 5 - 4

y = (x - 2)2 + 1

In this form we see that the graph can be obtained by translating the graph y = x2 right 2 units because of the x - 2, and up 1 units because of the +1.

y = x2 - 4x + 5 Reflections

(-x, y) is the reflection of (x, y) about y-axis

(x, -y) is the reflection of (x, y) about x-axis

Graphs of y = f(x) and y = f(-x) are reflections of one another about the y-axis

Graphs of y = f(x) and y = -f(x) are reflections of one another about the x-axis

The graph can be obtained by a reflection and a translation:

- Draw a graph of - Reflect it about the y-axis to get graph of - Translate this graph right by 2 units to get graph of Here is the graph of  If f(x) is multiplied by a positive constant c

The graph of f(x) is compressed vertically if 0 < c < 1

The graph of f(x) is stretched vertically if c > 1 The curve is not the graph of y = f(x) for any function f

Contact email: 