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Weeks (f) 0 1 2 3 4 5 6
Weight in grams (iv) 50 68 81 112 151 214 302
Solution. A graph is drawn in the figure showing the weight w of the marrow at the end of week t. To find the rate of increase of the weight of the marrow at the end of week 4, a tangent is drawn to the graph at that point P. The right-angled triangle ABC is completed, with AC being a suitable whole number of units long, which in this case happens to be 4 weeks. The length of BC is now required, and this appears to be 200 grams. Write
Gradient of AB =-------
200 grams 4 weeks
= 50 grams per week The rate at which the marrow is growing at the end of day 4 is 50 grams per week.
References: Gradient, Graphs, Rate of Change, Tangent.
The gram (abbreviation g) is a small unit of mass; it is one-thousandth of 1 kilogram (kg):
1000 g = 1 kg
An important relation is that 1 cubic centimeter of water (1 cm3) weighs 1 g, provided the water is at a temperature of 4° Celsius (1 cm3 is the same as 1 milliliter). Thus one can also say that 1 cubic meter (1 m3) of water weighs 1000 kg, or 1 tonne (abbreviation t; note that 11 = 1000 kg = 2200 pounds, whereas 1 U.S. ton = 2000 pounds).
References: CGS System of Units, Kilogram, Metric Units.
For information on statistical graphs see the following entries:
♦ Bar Graph
♦ Box and Whisker Graph
♦ Pie Graph
♦ Stem and Leaf Graph
♦ Frequency Curve
♦ Line Graph
♦ Scatter Diagram
♦ Cumulative Frequency Graph
♦ Time Series
♦ Normal Curve
For information on the transformations of the parabola, see the entry Quadratic Graphs. For the following basic curves and how to sketch them, see the following entries:
♦ Cubic graphs (Quadratic Graphs)
♦ Distance-time graph (Travel Graphs)
♦ Ellipse (Ellipse, also Conic Sections)
♦ Exponential or growth curve (Exponential Curve; see also the present entry)
♦ Hyperbola (Hyperbola; see also Conic Sections)
♦ Inequalities of Graphs (Inequality)
♦ Linear graphs straight-line graphs (Gradient-Intercept Form)
♦ Logarithmic curve (Logarithmic Curve, also Exponential Curve)
♦ Parabola (Quadratic Graph; see also Conic Sections)
♦ Qualitative graphs (Qualitative Graphs)
♦ Trigonometric curves (Trigonometric Graphs; see also Circular Functions)
♦ Velocity-time graphs (Acceleration)
♦ Arrow graphs (Arrow Graph)
In the present entry we shall study the following:
1. How to draw algebraic graphs (which are graphs drawn from algebraic equations) using a table of values.
2. How to draw graphs by sketching.
3. Conversion graphs (see also the entries Conversion and Extrapolation)
4. Circle graphs (see also the entry Conic Sections)
5. Using graphs to solve algebraic equations.
How to Draw Algebraic Graphs An algebraic graph is made up of a set of coordinates plotted on a Cartesian plane and joined by a straight line or by a curve. The set of coordinates is usually written as a table of values, and the coordinates are obtained by substituting into an algebraic equation. This method of drawing graphs is the most basic method and may be used whenever we have little or no idea of what kind of graph will result from the given algebraic equation.
Example 1. Draw the graph of the equation y = (x — 3)2, using values of x from 0 to 6.
Solution. The values of x = 0, 1, 2, 3, 4, 5, 6 are entered in a table of values as shown. It is convenient to break down the expression (x — 3)2 into (x — 3) and then square it.
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 arex = —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 -3-2-1012 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).
How to Draw Graphs by Sketching To leam 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.