Numerical View of Lines
and their Equations
In this lesson we see numerically how plotting points (x,y) satisfying an equation y = mx + b in which m and b are constant give points on a straight line in the plane. We further see that increasing x by 1 increases y by m. Moreover if m = b/a then increasing x by a increases y by ma = (b/a)a = b.
What is a Parameter? The role of m and b as parameter is also explained below.
First Numerical Example
For the equation y = 2x +4 of a line, we find and plot several ordered pairs (x,y) which satisfy the equation.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 2 | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
Observe y changes by 2 when x increases by 1. So the rise over run ratio (the slope) m =2 = coefficient of x in the equation y = 2x +4
These points are plotted below.
Then we joined the dots by drawing straight lines between them (linear interpolation). The geometrical reasoning met earlier implies these line segments belong to a straight liney = 2x +4 that we draw over the line segments and extend beyond in two directions, up and down.
In our drawing of straight lines associated with an equation y = mx +b, we assume plotting two to several points allows the full line or good portion of it to be drawn with the aid of a rule or straight edge passing through the points.
Rule of Thumb (1): When you are asked to graph an equation y = mx +b of a straight line, locate the x and y intercepts , and if they are different, draw a straight line through them. In the case b= 0, the straight line passes through the origin, and the x- and y-intercepts are both at the origin. To get a second point to draw your straight line, pick a value for x and compute y at that value. Numerical experience from several examples which you have done yourself or seen in a book may guide you in the selection of that second point.
The x- and y-intercept can be directly found by solving the equation of a line, here y = mx +b in two cases (i) for x given y =0 and (ii) for y given x = 0. The second case is easy. The algebraic methods works in general while numerical inspection methods may work, but they do so apart from skill development (and display) in algebra.Rule of Thumb (2): A straight line can be drawn through a pair of points, but errors in drawing the line become larger for points further away from the given pair. To lessen the errors in drawing that come from missing the center of the given dots that represent the points can be minimized by choosing the points far apart over the interval of interest. When x and y intercepts are not far apart in interval of interest, find and use another pair of points for drawing your straight line.
Second Numerical Example
For the equation y = -(¾)x + 1¼ of a line, we find and plot several ordered pairs (x,y) which satisfy the equation.
| x | -12 | -8 | -4 | 0 | 4 | 8 | 12 |
| y | 10¼ | 7¼ | 4¼ | 1¼ | -1¾ | -4¾ | -7¾ |
Observe
- x = 0 gives us the y-intercept, here y = 1¼.
- y changes by 4(¾) = -3 when x increases by 4.
- y changes by -4(¾) = 3 when x decreases by 4.
The last two rules could be used to fill in the table starting with the greatest, least or in the middle value of x, x = 0 for instance. Exact calculations with fractions are recommended as a skill building and displaying exercise for algebra.
plotted points and a line through them.
Since y changes by 4(¾) = -3 when x increases by 4, we see that the rise over run ratio, the slope or m = rise/run = -3/4 = - (¾) = the coefficient of x in y = -(¾)x + 1¼
Our selection of values for x does locate the x-intercept. But the table of values suggest y will be zero between 0 and 4.
To find the x-intercept exactly put replace y by zero in the equation y = -(¾)x + 1¼ to get the equation
0 = -(¾)x + 1¼
All the coefficients are multiples of ¼. So let multiple by its reciprocal, here 4, to eliminate fractions. That yields the equation
0 = -3x + 5
which in turn gives x = 3/5 when y = 0. That provides the exact location of the x-intercept.
Rule of Thumb (3): In plotting y = mx + b, when m = c/d is given by a fraction with small whole numbers in the numerator and denominator, select values of x of the form Nd. Then the y values will have the form Nc + b and increasing x by d will change y by c.
Remark: You need to be able to plot points along a line y = mx + b with b nonzero, but the interval of x values you select for the plot should include the x-intercept and y-intercept at say a quarter of length of the interval from either end. In plotting a straight line in mathematics, the main features of the line, it intercepts, are usually of the most interest.
In general, we follow the cosmetic convention in graphing that the main features should occupy the central region of the graph. What are the main features depends on the context provided by the current math topic or current application of mathematics.
Exercise 1: Draw the graph of the equation y = -4x +8 as follows. First find the x and y intercepts. Then choose an x-interval so that the x-coordinate of the x and y are about a quarter of the interval length from the interval endpoints. Then with the aid of a straight-edge or ruler, draw a straight-line over that x-interval through the x and y intercepts. Finally, label the x- and y-axes and points along it in accordance with the cosmetic conventions of your current mathematics class.
Exercise 2: Plot several points of the straight line y = 2x - 1.5
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Graphing: Two points are usually needed to draw a straight line. Use the x- and y- intercepts if the line does not pass through the origin. For best results (greatest accuracy) in drawing a line, take two points far apart. One point is enough is the line is horizontal or vertical. Label the horizontal and vertical axes with their names and coordinates. |
Parameters for Lines and Their Equations
Above we have met and graphed the following equations to obtain lines in the xy-coordinate plane.
- y = 2x +4
- y = -(¾)x + 1¼
- y = -4x +8
- y = 2x - 1.5
In these equations for straight lines, the variable y depends on the variable x. Now in each dependency has the form
y = mx + b
where m = the coefficient of x and
where b = the constant term = the value of y when x = 0. In each example, we may
obtain the value of a and b by inspection.
| y = mx+b | m | b |
| y = 2x +4 | 2 | 4 |
| y = -(¾)x + 1¼ | -¾ | 1¼ |
| y = -4x +8 | -4 | 8 |
| y = 2x - 1.5 | 2 | -1.5 |
In each example, we graph the variable y versus the variable x while m and b which we call the coefficient of x and the constant term remain constant. While m and b are constant along each row, that is, in each example, their values vary between examples. But we will not call them variables. We will call them parameters instead. These parameters does not vary in each example, it varies between examples.
Remark: Words have missing in the introduction of the algebraic shorthand roles of letters and symbols. In mathematics, we use letters and symbols to denote numbers and quantities. Those numbers and quantities may be known or not, constant in one sense or another, variable in one way or not, The mathematical habit of saying a letter in mathematics indicates a variable is not alway true. For example, the letter p is used to denote a constant. We can talk about and describe numbers and quantities. The adjectives or descriptiosn that apply to numbers and quantities can also be applied to the letters or symbols that stand for those numbers and quantities in our formulas and reasoning. To learn more, see chapters 8 to 12 in Volume 2, Three Skills for Algebra.