The absolute value parent function is a basic mathematical function that is written as:

$f(x) = |x|$

Key characteristics of the absolute value parent function:

1. Domain: The domain is all real numbers, $\mathbb{R}$, because you can input any real number into the function. $\text{Domain: } (-\infty, \infty)$

2. Range: The range includes all non-negative real numbers, because the absolute value is always zero or positive. $\text{Range: } [0, \infty)$

3. Shape: The graph forms a "V" shape, with the vertex at the origin (0,0). The graph is symmetrical about the y-axis.

4. Slope: For $x \geq 0$, the slope is 1, and for $x < 0$, the slope is -1.

5. Intercept: The function intersects the y-axis at $y = 0$, because $|0| = 0$.

Piecewise representation:

\begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ This function essentially measures the distance of any number from zero on the number line. --- Would you like more details or have any questions about this? Here are 5 related questions to explore: 1. How do transformations affect the absolute value parent function? 2. What happens when the absolute value function is shifted horizontally or vertically? 3. How do you find the vertex of a transformed absolute value function? 4. What is the real-world significance of the absolute value function? 5. How is the absolute value function related to piecewise functions? **Tip:** The absolute value function is often used to model situations where only the magnitude matters, such as distance or error.