
Linear functions are any functions that can be expressed in the form of | |
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where m and b are constants. |
Example 7
1. | 2x - 3y = 1 is a linear function. |
2. | y = 2 is a special linear function called a constant function. |
3. | x = -1 is linear but not a function. |
4. | y = x² - 3 is a function but not linear. |
To sketch the graph of a linear function, you just need to locate any two points on the graph and then draw a straight line passing through them. This, of course, requires an understanding that the graph of a linear function is a straight line.
A line is defined when we have sufficient information to derive its equation. The next reading discusses various ways to define a line. Note that in the discussion (x,y) represents a general point, and (x1,y1) represents a specific point on the line.
Reading
Read the following sections from an introduction to
linear functions from MIT.
http://dedekind.mit.edu/~djk/calculus_beginners/chapter03/section02.html
http://dedekind.mit.edu/~djk/calculus_beginners/chapter03/section03.html
Be sure to try out the applets built into these pages that allow you to experiment with graphing linear functions.
Note that the slope of a vertical line is undefined - not infinity. Slope is an important measure, but is not sufficient to define a line by itself. There are lines of which the slopes are the same.
Example 8
Given that:
Determine which of these lines are:
1. | parallel; or |
2. | perpendicular. |
Solution
Comparing the above equations with the point-slope equation we obtain
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Slope |
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1. | Since the slopes are all different, none of the lines are parallel. |
2. | Since the product of Slope(L1) and Slope(L4) is -1, L1 and L4are perpendicular. |
(Note that Slope(L2) × Slope(L3) = 1, not -1, and therefore L2 and L3 are not perpendicular.) |
Definition
The equation
is the two-point equation of the line that passes through the points (x1,y1) and (x2,y2).
Example 9
Find the equation of the line with x-intercept 2 and y-intercept -1. (Recall that the x-intercept is the x-coordinate where the graph and the x-axis intersect. The y-intercept is the y-coordinate where the graph and the y-axis intersect.)
Solution
The two intercepts imply that (2,0) and (0,-1) are two points on the line. Applying the two-point equation we have
Simplifying the equation gives
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