Consider
the function y = 2x. Compute
y for several values of x and set up the table:Graphical representation of the function y = mx
Consider
the function y = 2x. Compute
y for several values of x and set up the table:
x = . . . +2 +1 0
-1 - 2 ....
y = . . . +4 +2 0 -2 - 4 ....
Every corresponding pair of numbers is a pair of coordinates. Draw axes on graph paper and fix the units. As a rule, unless said especially, you use the same units on both axes. x = +1, y = +2 yields the point A, x = -1, y = -2 the point B, etc. You can construct in this manner as many points as you like; there are infinitely many of them.
All points, the coordinates of which satisfy a functional equation, yield the graph of the function.
You see that in this case all the points lie on a line. You speak of the equation of the straight line: y = 2x, or, more briefly, of the line y = 2x.
Away from the line, there exist no points the ordinates of which are twice their abscissae. Once you have recognized the graph of a functional equation, you can also extract from it new pairs of values, which satisfy the functional equation. For x = 1½, the graph yields y = 3, for x = 0.4, y = 0.8, etc.
Task: Let the ordinates be three times the
abscissae. Find the functional equation, a table of values and
its graph!
x = . . . +2 +1 0
-1 - 2 ....
y = . . . +6 +3 0 -3 - 6 ....
The functional equation is y = 3x.
Again, all the points lie on a straight line.You can repeat these actions for any equation of the form y = mx and always arrive at the same conclusion.
The image of y = mx is a straight line.
Task: The ordinates are to be one third of the abscissae.What is the functional equation? How is the line placed with respect to y = 2x and y = 3x?
Answer
Since the pair of the values x = 0, y = 0 satisfies every equation y = mx, you conclude that:
Every straight line y = mx passes through the origin.
In order to draw such a line, you require only two points, one of which can be the origin.
Task: Draw the line y =
4x.
For x = 1, you find the point A for which y = 4. Now draw the line through O and A and obtain the required straight line.
Exercises:
1.Draw the lines y = x and y = -x, y = 2, y = -2, x = 2, x = -2. It is a square with sides 4 with its diagonals!
2. Draw the line y = x/3. How is it placed with respect to the line y = 2x?