parent function
[parent function|parent function] The parent function refers to the base form of a function before any transformations are applied.
Definition
parent function The parent function refers to the base form of a function before any transformations are applied. Horizontal and vertical shifts involve adding constants to the input or to the function itself. A stretch or compression occurs when the parent function is multiplied by a constant |a| > 0.
Mechanism
parent function The parent function f(x) = 2^x serves as the foundation for transformations. A vertical stretch by a factor of 3 results in g(x) = 3(2^x), while a vertical compression by a factor of 1/3 yields h(x) = (1/3)(2^x). These transformations demonstrate how changes to the function's parameters affect its graph.
Comparison
parent function The parent function exhibits increasing behavior when b = 10, which is greater than one. This indicates that the function's rate of change is higher compared to scenarios where b is less than one. The relationship between b's value and the function's trend is directly tied to its increasing nature. Since b = 10 exceeds the threshold of one, the parent function demonstrates a clear upward trajectory. The evidence confirms that the parent function's increasing characteristic is contingent upon b being greater than one.
Applications
parent function Graphing the parent function f ( x ) = 2 x serves as a starting point for demonstrating transformations. A stretch with a = 3 produces g ( x ) = 3 ( 2 ) x , illustrated on the left in [link] . A compression with a = 1 3 results in h ( x ) = 1 3 ( 2 ) x , shown on the right in [link] .
Examples
parent function Graphing the parent function f ( x ) = 2 x serves as a starting point. Applying a stretch with a = 3 produces g ( x ) = 3 ( 2 ) x , demonstrated on the left in [link]. Using a compression factor of a = 1 3 results in h ( x ) = 1 3 ( 2 ) x , shown on the right in [link].
Adding Constant
parent function Adding constants to the input or the function itself results in horizontal and vertical shifts. These transformations differ from stretches/compressions, which involve multiplying the parent function f(x) = b x by |a| > 0. The parent function remains central to these changes.
Vertical Shift
parent function Vertical shift refers to the movement of a parent function up or down by adding a constant to the function itself. This differs from horizontal shifts, which involve adding constants to the input. Both types of shifts are distinct from stretches or compressions, which result from multiplying the parent function by a constant |a| > 0.