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  Concept of Lines in the Plane

Lines in the Plane:

Slope of a Line:

The slope is symbolized by the letter m.

Slope of a non-vertical line is stated in many ways. It is a rise above the run. It is the change in y over the change in x.

For two points (x1,y1) and (x2,y2) where x1≠ x2, the slope is  as m = (y2 - y1)/(x2 - x1)

When the slope is positive, then the line mounts from left to right. And when the slope is negative, the line drops from left to right. When the slope is zero, then the line is horizontal. When the slope is undefined, then the line is vertical.

Point-Slope Form of a Line:

The equation of non-vertical line passing via the points (x1, y1) and (x2, y2) and containing a slope m is specified by the equation:
y - y1 = m (x - x1)

The point which you call point 1 and the point which you call point 2 doesn’t matter.

We almost never leave equation of a line in point-slope form; however use it as a stepping ground to the final answer. The only exception here is whenever we are finding the asymptotes to hyperbolas in conic sections.

Historical Note:

Linear Interpolation is employing the equation of a line to estimate a value that falls between the two known points. Linear Interpolation is frequently used whenever looking up values in the table, and the value you require is not in the table; however between the two values that are in table. In good old days before there were calculators, we utilized linear interpolation to determine logarithms and trigonometric values. Now, the calculators contain those functions built into them, therefore there are fewer requirements to use interpolation. The Linear Interpolation is still utilized some in statistics; however for most of the things, we just utilize the value from the table that is closer or less likely to cause a much serious error.

Linear Extrapolation is the procedure of employing the equation of a line to estimate a value that falls outside the two known points.

Slope-Intercept Form of a Line:

The equation of non-vertical line crossing the y-axis at point (0, b) and containing slope m is provided by the equation:

y = m x + b

The point-slope form can be positioned into the slope-intercept form with little algebra.

The slope-intercept form of a line is what should be placed to the calculator to get it to the graph line.

General Form of a Line:

In common, the general form of anything will be the form where all constants and variables are on the left side of equation, in decreasing degree of the terms and alphabetically for such terms which have similar degree.

For a line, which means ax + by + c = 0

a and b cannot both be zero, when they were, then you have c = 0, that is a constant, not a linear function.

Vertical and Horizontal Lines:

Vertical lines are lines which have all the x-coordinates similar. Therefore the equation of a vertical line is x = a (where a is common abscissa).

Horizontal lines are the lines which contain all the y-coordinates similar. Therefore, the equation of a horizontal line is y = b (where b is common ordinate).

Parallel Lines:

Parallel lines are the lines in similar plane which do not intersect. The slope of parallel lines is similar.

Perpendicular Lines:

Perpendicular lines are the lines in similar plane which intersect at right angle. The product of non-vertical and non-horizontal perpendicular lines is a negative one. The other way of saying that is that the slopes of perpendicular lines are just opposite reciprocals of one other.

Calculator:

Employing the graphing calculator to the graph lines is fairly straightforward. Solve for y and press the expression into the calculator. Solving for y is equal to putting the equation to the slope-intercept form.

Be careful with the slopes which are in fractions while using the TI82 calculator.

Consider the given equations:

y = 1/2x + 2
y = x/2 + 2
y = (1/2)x + 2
y = 1/2*x + 2
y = 0.5x + 2
y = 0.5*x + 2

All the above equation except for the first one will give you the graph of line. The first one will provide you the graph of the rational function.

TI82 wrongly gives implied multiplication of a higher preference than division and supposes that whenever people place an expression devoid of a multiplication symbol after a division symbol, then they wish to encompass the whole expression in the denominator.

TI-82 takes y = 1/2x + 2 as y = 1/(2x) + 2, that is a rational function, not a line.

Several people use a slash (/) to point out division. You must use a horizontal division bar for the division. The slash is okay whenever there is no implied multiplication in denominator.

TI83 correctly handles the expression and all six expressions above will graph appropriately.

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