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


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