Overview
The graphs of linear functions can be transformed without changing the shape of the line by changing the location of the y intercept or the slope of the line. Those lines can be transformed by translation, rotation, or reflection, and still follow the slope-intercept form y = mx + b.
Family of Functions
The graphs of linear functions all have the same basic shape: a line. They form a family because they all follow that basic shape and their lines express the same equation, y = mx + b, the slope-intercept form. The most basic function in the family, f(x) =x, is called the parent function.
Translation
One line can be translated to another line if the y-intercept is changed. Since the slopes of the lines are still the same, but they cross the y axis at different points, they are parallel. Suppose one line has the equation y = x + 3 and another line has the equation y = x – 2. The slope of the line x is the same for both lines, but one crosses the y axis at 3, while the other crosses the y axis at -2. It is as if the entire line with the equation y = x + 3 slid down to fit in the position y = x – 2. The line y = x is parallel to the other two, as the slope x is still the same, but the y intercept is at the origin.
Rotation
Graphs of lines can also be changed so they have the same y intercept, but different slopes. The line may have a steeper slope or one that is less steep. It is as if the entire line were turned. For example, lines of direct variation have the same y-intercepts but different slopes. All the lines may have their y intercepts at the origin, but the line y = -3x is steeper than the line y = 2x, and the line y = x is not quite as steep.
Reflection
Remember that the reflection of a figure is its opposite across a line of symmetry. The reflection of a line is similar to the reflection of a figure, as it represents its mirror image. If a line has slope m, its reflection will have a slope multiplied by -1. Suppose a line can be represented by the equation y = 3x. Its reflection will have the slope (3) (-1), or -3.
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