# 1.2 - Lines in the Plane

## Slope of a Line

The slope is represented by the letter m.

The slope of a non-vertical line is defined several ways. It is the rise over 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 m = ( y2 - y1 ) / ( x2 - x1 )

If the slope is positive, the line rises from left to right. If the slope is negative, the line falls from left to right. If the slope is zero, the line is horizontal. If the slope is undefined, the line is vertical.

## Point-Slope Form of a Line

The equation of the non-vertical line passing through the points (x1,y1) and (x2,y2) and having slope m is given by the equation:

y - y1 = m ( x - x1 )

Which point you call point 1 and which point you call point 2 does not matter.

We almost never leave the equation of a line in point-slope form, but use it as a stepping ground to a final answer. One exception to this is when we're finding the asymptotes to hyperbolas in conic sections.

### Historical Note

Linear Interpolation is using the equation of a line to approximate a value which falls between two known points. Linear Interpolation is often used when looking up values in a table, and the value you need is not in the table, but between two values which are in the table. In the good old days before there were calculators, we used linear interpolation to find logarithms and trigonometric values. Now, the calculators have those functions built into them, so there is less need to use interpolation. Linear Interpolation is still used some in statistics, but for most things, we just use the value from the table which is closer or less likely to cause a more serious error (don't worry if that didn't make sense, take a statistics course, and it will).

Linear Extrapolation is the process of using the equation of a line to approximate a value which falls outside two known points.

## Slope-Intercept Form of a Line

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

```   y = m x + b
```

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

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

## General Form of a Line

In general, the general form of anything will be the form where all the variables and constants are on the left side of the equation, in decreasing degree of the terms and alphabetically for those terms that have the same degree.

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

a and b can not both be zero, if they were, then you would have c=0, which is a constant, not a linear function.

## Vertical and Horizontal Lines

Vertical lines are lines that have all the x-coordinates the same. So, the equation of a vertical line is x=a (where a is that common abscissa).

Horizontal lines are lines that have all the y-coordinates the same. So, the equation of a horizontal line is y=b (where b is that common ordinate).

## Parallel Lines

Parallel lines are lines in the same plane that do not intersect. The slope of parallel lines is the same.

## Perpendicular Lines

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

## Calculator

Using the graphing calculator to graph lines is pretty straightforward. Solve for y and enter the expression into the calculator. Solving for y is equivalent to putting the equation into slope-intercept form.

One warning though for users of the TI82 calculator. Be careful with slopes that are fractions.

Consider the following 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 of the above except for the first one will give you the graph of a line. The first one will give you the graph of a rational function.

The TI82 incorrectly gives implied multiplication a higher preference than division and assumes that when people put an expression without a multiplication symbol after a division symbol, they wish to have the whole expression in the denominator.

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

Some people use a slash (/) to indicate division. You should use a horizontal division bar for division. A slash is okay when there is no implied multiplication in the denominator.

The TI83 properly handles the expression and all six expressions above will graph properly.