Dear graphpaper2,
My tutorial on graph paper follows.
Graph paper is ruled with horizontal and vertical lines that form a
square grid. The side of each small square is usually a half-centimeter
or a quarter-inch long, but it isn't necessary to know exactly what this
length is. When we use graph paper, we simply say that each square is
a unit square. The length of each side of a unit square is called the
unit length.
Whenever we draw an object on the graph paper, we do so using a scale
that tells us what distance the unit length represents. There is no
fixed rule as to what the scale should be. It can vary from diagram to
diagram, but it should always be the same within a single diagram. We
should try to choose a scale that will let us draw the object of interest
at a reasonable size.
For example, suppose we want to use a sheet of graph paper to make
a diagram of Farmer Jones' lettuce field. The lettuce field is a
rectangle 60 meters long from top to bottom and 90 meters wide from side
to side. Let us first graph the field using a scale in which the unit
length represents 30 meters.
Question: How long and how wide is Farmer Jones' lettuce field in terms
of 30-meter units?
Answer: The lettuce field is 60/30 = 2 units long and 90/30 = 3 units
wide.
To graph the lettuce field, start by drawing a vertical line two units
long. The grid will help you make the exact measurement. Your line
should start at one corner of a square and cover two edges before
ending at another corner. Let this be the left border of the lettuce
field. Now, starting at the upper end of this line, draw a horizontal
line that extends toward the right and is exactly three units long. This
is the top border of the field. From its right end, draw a vertical line
extending downward and exactly two units long. This is the right border,
and now you can complete the diagram by drawing the bottom border.
We end up with a two-by-three-unit rectangle to represent Farmer Jones'
60-by-90-meter field. It's not a very large diagram, is it? If we
now wanted to draw, say, Farmer Smith's 10-by-10-meter beet field for
comparison, we would end up something almost too small to see. So let's
draw another diagram of Farmer Jones' lettuce field, this time using a
scale of 10 meters to the unit. Once again, follow the grid lines and
make sure all borders meet at the corners. You should end up with a
rectangle six units tall and nine units wide.
Now use the same scale to draw Farmer Smith's 10-by-10-meter beet
field. Place the diagrams side by side so that you can compare their
relative sizes on the graph paper.
Question: What kind of trouble would we run into if the dimensions of
Farmer Smith's beet field were 15 meters by 10 meters?
Answer: On a scale of 10 meters to the unit, a 15-meter line would span
one and a half unit squares. This would make it impossible to draw all
the borders along the grid lines of the graph paper.
We can also use graph paper to draw more interesting shapes. For example,
Farmer Murphy's artichoke field is in the shape of a triangle whose
borders are 60 meters, 80 meters, and 100 meters long. Draw this triangle
on a scale of 20 meters to the unit, which means that the diagram will
have edges of length 3, 4, and 5 units respectively. Start by drawing
the 4-unit edge lying horizontally at bottom. From the left end, draw
the 3-unit border upward. Finally, connect the top end of the left border
with the right end of the bottom border.
You can use a ruler to verify that the final edge is 5 units long. For
example, if the unit squares of your graph paper have half-centimeter
sides, the longest edge will be 5*0.5 = 2.5 centimeters long. If the
paper is ruled with a quarter-inch grid, the longest edge must have a
length of 5*0.25 = 1.25 inches.
Next, let's use graph paper to investigate the relationship between
length and area. Consider a unit square, which is just the size of the
diagram you made of Farmer Smith's beet field. This is a square whose
sides are each one unit long, so we say that it covers an area of one
square unit. Now draw a square whose sides are each two units long. Count
the number of squares lying within this square: there are four. So the
2-by-2 square has an area of four unit squares. Next, do the same for a
3-by-3 square and a 4-by-4 square. Now draw a 1-by-2 rectangle, a 2-by-3
rectangle, and a 2-by-4 rectangle, counting the squares inside each to
compute its area.
Question: What is the relationship between the area of a rectangle and
the lengths of its edges?
Answer: The area of a rectangle is the product of its height and width.
You have probably noticed that by doubling the side length of a square
or rectangle, we quadruple its area. Let us see whether this principle
holds for other shapes as well. Remember the diagram you drew of Farmer
Murphy's artichoke field? This is a triangle with edges 3, 4, and 5 units
long. We can't determine its area by counting how many unit squares it
covers, since the longest edge slices through some of the squares. But
we can still compute the area of the triangle with the following trick.
Observe that the triangle forms exactly half of a rectangle 3 units
high and 4 units wide. Thus, its area must be exactly half that of the
rectangle. Since the rectangle has an area of 3*4 = 12 square units,
the triangle must have an area of 12/2 = 6 square units. Now draw a
triangle with twice the side length, meaning that it has edges 6, 8,
and 10 units long. What is the area of this triangle? Do the same for
a triangle with edges 9, 12, and 15 units in length.
Question: If we double the length of each edge of a triangle, what
happens to its area?
Answer: As with rectangles, its area quadruples.
To express the relationship between the area of a square and the length
of its sides, we can use the equation y = x^2, pronounced "y equals x
squared". This says that if the side length of a square is x, then it
covers an area of y square units, where y is the same as x times x. The
expression "x squared" means the same as "x times x", and x^2 is just a
convenient way of writing it. Since the value of y is strictly determined
by the value of x, this equation is what is known as a function. The
function y = x^2 defines a mapping from the independent variable x to
the dependent variable y.
An interesting way to investigate the properties of a function is to
plot it on a grid. The resulting diagram is called a graph, hence the
name graph paper. To draw a graph, we first have to decide on a scale
and a range. You are already familiar with the notion of a scale: it
tells us what distance is represented by each unit length. The range of
a graph is what determines how much of it and what part of it we want to
see. As with the scale, there is no fixed rule for deciding the range
of a graph. It may take some trial and error to arrive at a good range
for a graph in a given situation.
For the time being, let us draw a graph for y = x^2 on a one-to-one scale,
with x values ranging from 0 to 5. To determine the corresponding range
of y values, we make a chart showing the value of x^2 for sample values
of x from smallest to biggest.
x x^2
--- ---
0 0
1 1
2 4
3 9
4 16
5 25
So if x ranges from 0 to 5, y will range from 0 to 25. On a 1:1 scale,
this requires that we draw a horizontal axis 5 units long and a vertical
axis 25 units tall. The axes of a graph are just a pair of lines showing
the ranges of the x and y variables. The horizontal axis is also known
as the x-axis, and the vertical as the y-axis.
Begin by drawing a 25-unit-long vertical line near the left edge of
the paper, then connect to its bottom end, extending to the right,
a horizontal line 5 units long. Draw an arrowhead at the top end of
the vertical axis and the right end of the horizontal axis to show the
direction of increasing magnitude. Also, in order to show the range and
scale, mark the distance along each axis at 2-unit intervals. The point
where the axes meet, called the origin, is distance 0 on each axis. You
should end up with the markings 0, 2, 4 on the horizontal axis, and
markings 0, 2, 4, 6, ... 20, 22, 24 on the vertical.
You must now plot some function points. In the chart above, we have six
such points, which are just pairs of x,y values. For each pair, locate
the x value along the horizontal axis and consider the vertical grid line
extending from it. Now find the corresponding y value on the vertical
axis, and look at the grid line extending horizontally from there. The
intersection of the vertical line extending from the horizontal axis
with the horizontal line extending from the vertical axis is a function
point. Mark it with a small black circle, and do the same for all six
function points.
Finally, connect the points in order from right to left. You should
do this freehand, without a ruler, to emphasize the curvature of the
graph. The graph rises more and more sharply as it extends to the right,
showing that the function y = x^2 increases at an ever greater rate as
x grows in the positive direction.
Question: Can you draw a 1:1 scale graph of y = x^2 for x values ranging
from 0 to 10?
Answer: Probably not. The corresponding y values range from 0 to 100,
so you would need a sheet of graph paper at least 100 squares high. You
could draw the graph on a smaller scale, however.
Next, draw a 1:1 scale graph of y = x^2 for x values ranging from -5
to 5. In this case, the vertical axis will rise from the middle of the
horizontal axis. Although the x range has doubled in size, the y range
will be the same as before. Do you see why? You can be confident that
you have plotted the graph properly if it ends up being symmetrical in
the vertical axis. This means that the left half of the graph is the
mirror image of its right half.
It has been a pleasure to address this question on your behalf. If you
feel that any part of my answer requires correction or elaboration,
please let me know through a Clarification Request so that I have a
chance to fully meet your needs before you assign a rating.
Regards,
leapinglizard |
