The function $y = |2 - 3x| - 4x^2$ combines absolute value and a quadratic term. Let's analyze it step by step:

1. Behavior of the Absolute Value:

• The term $|2 - 3x|$ represents an absolute value function. It behaves differently depending on whether $2 - 3x$ is positive or negative:

  • If $2 - 3x \geq 0$, then $|2 - 3x| = 2 - 3x$.
  • If $2 - 3x < 0$, then $|2 - 3x| = -(2 - 3x) = -2 + 3x$.

• The critical point occurs at $x = \frac{2}{3}$, where $2 - 3x = 0$.

2. Quadratic Term:

• The term $-4x^2$ is a downward-opening parabola (negative coefficient of $x^2$).

3. Piecewise Representation:

The function can be expressed piecewise based on the absolute value behavior:

\begin{cases} (2 - 3x) - 4x^2 & \text{if } x \leq \frac{2}{3} \\ (-2 + 3x) - 4x^2 & \text{if } x > \frac{2}{3} \end{cases}$$ ### 4. **Key Features**: - **Vertex**: The parabola and the absolute value contribute differently. To find specific critical points and local extrema, evaluate derivatives or plot key points. - **Symmetry**: The absolute value introduces non-smooth behavior at $$x = \frac{2}{3}$$. - **Domain**: The function is defined for all real $$x$$ ($$\mathbb{R}$$). - **Range**: Depends on analysis of critical points and behavior as $$x \to \pm \infty$$. Would you like me to compute the derivative for further analysis, plot the function, or identify specific points like maximum/minimum? Let me know! --- #### 5 Questions to Expand: 1. What is the domain and range of the function explicitly? 2. How does the graph of $$y = |2 - 3x| - 4x^2$$ behave at $$x = \frac{2}{3}$$? 3. Can we compute the derivative to identify critical points? 4. How would removing the quadratic term $$-4x^2$$ affect the function? 5. What is the significance of combining absolute values with quadratic functions in modeling? #### Tip: When graphing piecewise functions like this, always analyze critical points where the definition changes (e.g., $$x = \frac{2}{3}$$ here).