To find the transformation of a linear function, identify how the parent function, f(x) = x, has been shifted, stretched/compressed, or reflected. This involves analyzing the equation of the transformed function.
Understanding Linear Function Transformations
Linear function transformations involve altering the basic linear function, f(x) = x. These alterations can change the position, slope, and orientation of the line. Here's a breakdown:
1. Vertical Shifts (Translations)
- Upward Shift: Adding a constant k to the function, f(x) + k, shifts the graph upwards by k units. For example, f(x) = x + 3 shifts the line up 3 units.
- Downward Shift: Subtracting a constant k from the function, f(x) - k, shifts the graph downwards by k units. For example, f(x) = x - 2 shifts the line down 2 units.
2. Horizontal Shifts (Translations)
- Leftward Shift: Replacing x with (x + h) in the function, f(x + h), shifts the graph left by h units. For example, f(x) = (x + 4) shifts the line 4 units to the left.
- Rightward Shift: Replacing x with (x - h) in the function, f(x - h), shifts the graph right by h units. For example, f(x) = (x - 1) shifts the line 1 unit to the right.
3. Vertical Stretches and Compressions
- Vertical Stretch: Multiplying the function by a constant a where |a| > 1, af(x), vertically stretches the graph. For example, f(x) = 2x stretches the line vertically.
- Vertical Compression: Multiplying the function by a constant a where 0 < |a| < 1, af(x), vertically compresses the graph. For example, f(x) = 0.5x compresses the line vertically.
4. Reflections
- Reflection Across the x-axis: Multiplying the function by -1, -f(x), reflects the graph across the x-axis. For example, f(x) = -x reflects the line across the x-axis.
- Reflection Across the y-axis: Replacing x with -x, f(-x), reflects the graph across the y-axis. In the case of f(x) = x, the transformation is f(-x) = -x, which is the same as reflecting across the x-axis.
Combining Transformations
Often, linear functions undergo multiple transformations. A general form representing multiple transformations is:
g(x) = a(x - h) + k
Where:
- a represents vertical stretch/compression and reflection across the x-axis.
- h represents horizontal shift.
- k represents vertical shift.
Example:
Consider the function g(x) = 3(x + 2) - 1.
- The parent function, f(x) = x, has been:
- Vertically stretched by a factor of 3.
- Shifted 2 units to the left.
- Shifted 1 unit down.
By carefully analyzing the equation of the transformed linear function, you can identify each individual transformation and describe the overall change from the parent function.
