The given equation is:

\[
y = 2x^3 - 5
\]

This is a cubic function. Let's break down some important features:

1. **Type of Function**: It is a cubic function because the highest power of \(x\) is 3.
2. **Intercepts**:
   - **Y-intercept**: The point where \(x = 0\), which is \(y = 2(0)^3 - 5 = -5\). So, the y-intercept is \((0, -5)\).
   - **X-intercept**: To find the x-intercept(s), set \(y = 0\) and solve for \(x\):
     \[
     0 = 2x^3 - 5 \quad \Rightarrow \quad x^3 = \frac{5}{2} \quad \Rightarrow \quad x = \sqrt[3]{\frac{5}{2}}
     \]
     So, the x-intercept is \(x = \sqrt[3]{\frac{5}{2}}\), which is approximately \(x \approx 1.3572\).
3. **Slope/Rate of Change**: Since it's a cubic function, the slope is not constant. The slope increases as \(x\) increases because the derivative is positive for large values of \(x\).
4. **Derivative**: To understand the rate of change or the slope at any point, we can take the derivative:
   \[
   \frac{dy}{dx} = 6x^2
   \]
   This shows that the slope is positive for all \(x\), except when \(x = 0\), where the slope is 0 (indicating a critical point).
5. **Critical Points and Behavior**: There is a critical point at \(x = 0\) because the derivative equals 0 there. Since the slope changes from negative to positive as we pass through \(x = 0\), this point represents a local minimum.

Would you like further details on any specific aspect of this function?

Here are 5 questions to explore further:
1. How do we find the second derivative and what does it tell us about the function's concavity?
2. Can you explain how to sketch the graph of this cubic function?
3. What happens as \(x\) approaches positive and negative infinity in this function?
4. Can you calculate the inflection points for the function \(y = 2x^3 - 5\)?
5. How would we solve the equation \(2x^3 - 5 = 0\) exactly?

**Tip**: For cubic functions, understanding the derivative helps reveal crucial details about local extrema and points of inflection.

Explore the cubic function y = 2x³ - 5 with a detailed breakdown of its intercepts, critical points, and derivative. This problem covers key algebraic concepts such as finding the x-intercept, y-intercept, and calculating the derivative. Suitable for Grades 10-12.

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