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Direct variation occurs when 2 variables vary directly which means they are proportional. This is created by a constant "rate of change"(slope). (0, 0) must be a solution.
Some examples would be: y = 3x or y = (-1/2)x These are NOT direct variation relationships: y = 3x - 2 or y =(-1/2)x + 7 Linear Functions are related to direct variation but a little more general. Linear functions refer to any function whose graph is a line which can be expressed in the form y = mx + b where m and b are constants representing slope and y intercept. So...y = 3x - 2 above is Linear but not direct variation. Linear Functions may be expressed by using tables, equations, or graphs. The concept of slope must be examined when discussing Linear Functions...
The change is found by subtracting like coordinates, either provided or found through a table. In words, slope describes the steepness of the line. Slope can also be found by looking at "m" if the equation is of the form y = mx + b. Example (1): Find the slope of the line through points (7, 2) and (1, 1). Solution: (2 - 1)/(7 - 1)= 1/6 = 1/6 Example (2): Find the slope of the line through points (-3, 5) and (6, 2). Solution: (5 - 2)/(-3 - 6)= 3/-9 = -(1/3) Example (3): Find the slope of the line f(x) = -4x + 3 Solution: slope = -4 Example (4): Find the slope of the line y = 1.8 x - 4.4 Solution: slope = 1.8 Example (5): Using data from a previous lesson...
Solution: Choose any two data points, ie. (2, 8) and (4, 14) and substitute into the slope formula (8 - 14)/ (2 - 4) = -6/-2 = 3. The slope of the line described by the table of values is 3. Before we end the discussion of slope, we must consider 2 Special Cases: Horizontal and Vertical Lines. Since slope describes steepness, and a horizontal line is said to have "no" steepness, its slope = 0. Since a vertical line is so steep that no numerical value can be assigned to it, its slope is said to be undefined. Examples: Horizontal: y = (some constant) ("m" value is missing or equal to 0) Vertical: This should be the opposite of horizontal and thus we get x = (some constant) (looks very different from basic y = mx + b form) |