Question

Definition of a parent function

Answer 1

A parent function is the simplest form of a particular category of functions that retains the key characteristics of that category. It serves as a foundational model from which other more complex functions can be derived through transformations such as shifts, stretches, compressions, or reflections.

For example, here are some common types of parent functions:

1. **Linear Function**: \( f(x) = x \)
This is the simplest linear function, which produces a straight line.

2. **Quadratic Function**: \( f(x) = x^2 \)
This function generates a parabolic curve (U-shaped graph).

3. **Cubic Function**: \( f(x) = x^3 \)
This function produces an S-shaped curve.

4. **Absolute Value Function**: \( f(x) = |x| \)
This function creates a V-shaped graph.

5. **Exponential Function**: \( f(x) = a^x \) (where \( a > 0 \) and \( a \neq 1 \))
This function represents exponential growth or decay.

6. **Logarithmic Function**: \( f(x) = \log(x) \)
This function shows a growth rate that decreases over time.

7. **Trigonometric Functions**:
- Sine: \( f(x) = \sin(x) \)
- Cosine: \( f(x) = \cos(x) \)

Each of these functions represents a category of functions, and variations of these functions are often formed by applying transformations. For example, the function \( f(x) = x^2 + 3 \) is a transformation of the parent function \( f(x) = x^2 \). Understanding parent functions is foundational for analyzing and graphing more complex functions.