Write the equation for and sketch the graph of the parent (toolkit) function of 𝑓(𝑥)

The parent (toolkit) function refers to a set of basic, commonly used functions in mathematics, which serve as building blocks for more complex functions. These parent functions have well-defined equations and graphs. Without specifying a particular function, I will assume you're asking for some common examples of parent functions. Here are five examples with their equations and a description of their graphs:

1. **Linear Function**:
   - **Equation**: \( f(x) = x \)
   - **Graph**: A straight line passing through the origin with a slope of 1.

2. **Quadratic Function**:
   - **Equation**: \( f(x) = x^2 \)
   - **Graph**: A parabola that opens upward, with its vertex at the origin.

3. **Cubic Function**:
   - **Equation**: \( f(x) = x^3 \)
   - **Graph**: An S-shaped curve that passes through the origin, with symmetry about the origin.

4. **Absolute Value Function**:
   - **Equation**: \( f(x) = |x| \)
   - **Graph**: A V-shaped graph with its vertex at the origin, opening upwards.

5. **Square Root Function**:
   - **Equation**: \( f(x) = \sqrt{x} \)
   - **Graph**: A half-curve starting at the origin, increasing slowly as \( x \) increases.

If you have a specific parent function in mind, I can focus on that. Otherwise, I can sketch one of these graphs for you.

Would you like to see a graph of a particular one, or more details on any of these functions?

### Relative Questions:
1. What is the domain and range of the quadratic parent function \( f(x) = x^2 \)?
2. How does the graph of \( f(x) = |x| \) differ from \( f(x) = x^2 \)?
3. What transformations can be applied to the cubic function \( f(x) = x^3 \)?
4. What is the effect of a negative sign on the square root function \( f(x) = -\sqrt{x} \)?
5. How does the slope of the linear function change if we modify it to \( f(x) = 2x \)?

### Tip:
When transforming parent functions, changes like vertical shifts, horizontal shifts, stretches, and compressions are key to understanding how the graph will behave.

Write the equation for and sketch the graph of the parent (toolkit) function of 𝑓(𝑥).

Explore the equations and graphs of five common parent functions: linear, quadratic, cubic, absolute value, and square root functions. This guide provides insights into their characteristics and visual representations, useful for students in Grades 9-12 studying basic algebraic functions.

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