For the quadratic estimated equation  yˆ=b0+b1x+b2x^2, which of the following expressions must be zero to minimize or maximize the predicted y?

Multiple Choice
−b2 ÷ 2b1
−b1 ÷ 2b2
b1 + 2b2x
2b1 + b2x

To determine the expression that must be zero to minimize or maximize the predicted \( y \) in the quadratic equation \( \hat{y} = b_0 + b_1x + b_2x^2 \), we need to take the derivative of \( \hat{y} \) with respect to \( x \) and set it equal to zero.

The derivative of \( \hat{y} \) with respect to \( x \) is:

\[
\frac{d\hat{y}}{dx} = b_1 + 2b_2x
\]

To find the critical point (which could be a minimum or maximum), we set this derivative equal to zero:

\[
b_1 + 2b_2x = 0
\]

So, the expression that must be zero is:

\[
b_1 + 2b_2x = 0
\]

This corresponds to option **(3) \( b_1 + 2b_2x \)**.

Would you like more details or have any questions?

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Here are 8 related questions:

1. How do you determine if the critical point is a minimum or maximum?
2. What role does the sign of \( b_2 \) play in determining the shape of the quadratic curve?
3. Can there be multiple critical points in a quadratic equation?
4. How would you find the vertex of a quadratic function?
5. What happens to the quadratic curve if \( b_2 = 0 \)?
6. What are the implications of \( b_1 = 0 \) on the graph of the quadratic function?
7. How does changing \( b_0 \) affect the graph of the quadratic equation?
8. How do you solve a quadratic equation using the quadratic formula?

**Tip:** The vertex of a parabola given by \( \hat{y} = b_0 + b_1x + b_2x^2 \) occurs at \( x = -\frac{b_1}{2b_2} \).

Understand how to maximize or minimize the predicted y in a quadratic equation \( \hat{y} = b_0 + b_1x + b_2x^2 \). Learn the critical expression to set to zero for optimization. Suitable for high school students studying quadratic equations and derivatives.

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