Graphics of the general linear function y = mx + n

Task1: Graph y = 2x + 1.

There are different approaches to this task. As before, you can make a table and enter different points on graph paper, the result of which is shown in the figure.

Since we know the function y = mx quite well, we may compare y = 2x + 1 with y = 2x and find an alternative. We see that throughout y=2x+1 is greater by 1 than y = 2x, that is, that all points of the line y=2x must be shifted upwards by one unit parallel to the ordinate axis, as is shown in the next figure. In other words, y = 2x + 1 is parallel to y = 2x, that is, it is also a straight line.

This interpretation is independent of the values of m and n.

The Image of y=mx+n is a straight line.

For this reason, this function is called a linear function (lat. linea = straight line).

You obtain the straight lines y = x/2 + 3 and y = x/2 - 3 by parallel displacement of the line y=x/2 by n = + 3 or n = - 3 (cf. next figure).

Linear functions which differ only by their constant term yield parallels. n is the cut-off on the y-axis.

The parallel displacement tells you that m is also for the general linear function the inclination of the straight line; in the figure shown it is PQ/AQ.

Task 2: Draw the line y = x/2 + 3.

These rules now simplify the drawing of this line. You can either draw it by means of two points which you connect or you can draw y = x/2 and then construct its parallel at the ordinate 3. But you can also do the following:

The constant n yields the point A. If P is to be a point of the line, then PQ/AQ = m = 1/2. You can achieve this by moving right 2 units from A to Q and then from Q one unit upwards to P. AP is the required line.

You could also have moved 4 units to the right and then 2 units upwards; the only important aspect is that you must have PQ/AQ = m = 1/2.

Task 3. Does 2/x + 3/y = 4/xy - 2/x + 4/y represent a straight line?

Multiply both sides of the equation by xy:

The last equation has the form y = mx + n and is the equation of a straight line.

Not every equation between x and y can be transformed into the normal form y = mx + n. For example, x/y + 2/x = 3, on multiplication by xy, becomes x²+2y=3xy, an equation which contains x² as well as xy.

You need not always complete the transformation into the normal form. For example, you detact immediately in the equation (y - 3)·4 + (x - 5)·8 = (x - 4)·4 that expansion of the brackets yields only linear terms, so that the equation can be reduced to the form y = mx + n and therefore represents a straight line. For its construction, you employ two points the choice of which you make as simple and convenient as possible. In this problem, you set x = 4, then x = 5, because in this way you cause terms to disappear. You then draw the line with A(4,5) and B(5,4).

The equation y=0·x+b or y=b yields the same ordinate b for every abscissa x:

y = b represents the parallel to the abscissa at distance b.

The equation x=0·y+a or x=a yields the same abscissa a for every ordinate y:

x = a represents the parallel to the ordinate axis at distance a.

Exercises:

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