Final answer:
Transformations of linear functions involve shifting, reflecting, compressing, or stretching the graph of a function. These modifications include adding or subtracting a constant, multiplying by a factor, or reflecting the graph across an axis.
Step-by-step explanation:
The question relates to the transformation of linear functions, which is a mathematical concept dealing with alterations made to the graph of a linear function. These transformations include shifting (up or down), reflecting (across an axis), compressing (horizontally or vertically), and stretching (vertically). To understand these transformations, consider a simple linear function, such as f(x) = mx + b.
- Shift up: This refers to moving the entire graph of the function vertically upward in the coordinate system. In equation terms, if you add a constant k to f(x), the function becomes f(x) = mx + (b + k).
- Reflect across the x-axis: This transformation produces a mirror image of the graph across the x-axis. Mathematically, if you multiply the function by -1, the function becomes f(x) = -mx - b.
- Horizontal compression: This occurs when the graph becomes narrower along the x-axis. It can be achieved by multiplying the x variable by a factor greater than 1. If the factor is a, the function becomes f(x) = m(ax) + b.
- Vertical stretch: Opposite of compression, this makes the graph taller along the y-axis. If the stretch factor is c, you would multiply the entire function by c, leading to f(x) = c(mx) + cb.
To apply these transformations using a calculator, you input the linear equation and select the type of transformation you want to perform along with the corresponding value, such as the shift value, the factor of compression or stretch, or simply indicate a reflection.