The A to Z of Mathematics by Thomas H. Sidebotham

Author:Thomas H. Sidebotham [Sidebotham, Thomas H.]
Language: eng
Format: epub
Published: 2011-02-17T05:00:00+00:00

228

GRAPHS

The table of values is now completed.

x

0

1

2

3

4

5

6

( x − 3)

−3

−2

−1

0

1

2

3

y = ( x − 3)2

9

4

1

0

1

4

9

The coordinates ( x, y) to be plotted on the axes are (0, 9), (1, 4), (2, 1), (3, 0), (4, 1), (5, 4), and (6, 9), which are taken from the table of values. The x-axis is drawn from 0

to 6 and the y-axis from 0 to 9 (see figure a). These two axes form the Cartesian plane.

Since the range of y numbers is quite large compared to the range of x numbers, it is convenient to choose a scale of one square to 2 units for y. The graph then takes up less space. It is only necessary to label the y-axis every two units, 0, 2, 4, 6, 8.

When the points are plotted they are joined with a smooth curve, using a pencil. This curve is the graph of the algebraic equation y = ( x − 3)2, and is called a parabola, or quadratic graph.

Example 2. Sketch the graph of the equation y = 2− x for values of x from −3 to 3.

Solution. The values of x we use are x = −3, −2, −1, 0, 1, 2, 3 in the following table of values. The y values can be worked out using the exponent key yx on the calculator.

x

−3

−2

−1

0

1

2

3

y = 2− x (2 dp)

8

4

2

1

0.5

0.25

0.13

Plotting the coordinates (−3, 8), (−2, 4), (−1, 2), (0, 1), (1, 0.5), (2, 0.25), and (3, 0.13) on the axes gives the exponential graph y = 2− x (see figure b).

y

8

y

6

8

4

6

4

2

2

0

2

4

6 x

−2

0 1 2

x

−1

(a)

(b)

How to Draw Graphs by Sketching To learn how to draw sketch graphs of the basic curves, search for the curves by name (see the list at the beginning of this entry).

A general description is given here.

We can draw graphs accurately by making a table of values by substituting into an algebraic equation and then plotting the points on a Cartesian plane, as above. Or we can sketch the graphs of algebraic equations by having a good knowledge of their GRAPHS

229

general shape. This may involve finding their intercepts or doing transformations of the basic curves. This sketching of curves is usually a “snapshot” of the general shape and its properties, and usually does not give as accurate a result as drawing them by a table of values. The next example uses intercepts.

Example. Sketch the graph of the parabola y = ( x − 2)( x + 3).

Solution. We know that the basic shape of the parabola is as shown in the top part of figure c. A parabola can have up to two x intercepts and one y intercept. To find the x intercepts, substitute y = 0 in the equation of the curve, and solve the resulting quadratic equation:

0 = ( x − 2)( x + 3)

x = 2 and x = −3

See Solving a Quadratic Equation

y

2 x

−3

−6

(c)

To find the y intercept, substitute x = 0 in the equation of the curve: y = (0 − 2)(0 + 3)

y = −2 ×

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