Question

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Explanation:

Step1: Identify polynomial leading term

The given polynomial is $y = -0.000023x^3 + 0.0042x^2 + 0.75x$. The leading term is $-0.000023x^3$, with degree 3 (odd) and leading coefficient $-0.000023$ (negative).

Step2: Apply end behavior rules

For odd-degree polynomials with negative leading coefficients:

• As $x \to +\infty$, $y \to -\infty$
• As $x \to -\infty$, $y \to +\infty$

(Note: For this farming context, $x \geq 0$ since nitrogen application cannot be negative, so we only consider $x \to +\infty$ for practical use, but the mathematical end behavior includes both directions.)

1. Axes Labeling: Set the x-axis as "Nitrogen Application (pounds per acre)" and y-axis as "Corn Yield (bushels per acre)".
2. Key Features:

• y-intercept: Set $x=0$, so $y=0$. This is the point $(0,0)$, meaning 0 nitrogen gives 0 yield.
• x-intercepts: Solve $-0.000023x^3 + 0.0042x^2 + 0.75x = 0$. Factor out $x$: $x(-0.000023x^2 + 0.0042x + 0.75) = 0$. The roots are $x=0$, and using the quadratic formula on $-0.000023x^2 + 0.0042x + 0.75 = 0$:

$$x = \frac{-0.0042 \pm \sqrt{(0.0042)^2 - 4(-0.000023)(0.75)}}{2(-0.000023)}$$
Calculating gives one positive root $x \approx 213.4$ (the negative root is not meaningful for this context), so the x-intercepts are $(0,0)$ and $(213.4, 0)$.

• Maxima/Minima: Take the derivative $y' = -0.000069x^2 + 0.0084x + 0.75$, set to 0 and solve for $x$. Using the quadratic formula:

$$x = \frac{-0.0084 \pm \sqrt{(0.0084)^2 - 4(-0.000069)(0.75)}}{2(-0.000069)}$$
The positive root is $x \approx 164.5$, substitute back to find $y \approx 101.2$ (local maximum). The negative root is not meaningful. There is a local minimum at a negative $x$-value, which is not relevant to the farming context.

1. Graph: Plot these points, sketch the cubic curve that starts at $(0,0)$, rises to the local maximum $(164.5, 101.2)$, then falls to the x-intercept at $(213.4, 0)$ and continues downward as $x$ increases.

Answer:

As $x \to +\infty$, $y \to -\infty$; As $x \to -\infty$, $y \to +\infty$
(For the practical farming context where nitrogen application $x \geq 0$: As the amount of nitrogen applied per acre increases to very large values, the corn yield per acre decreases to negative values, which is not agriculturally meaningful, but this is the mathematical end behavior of the polynomial.)

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For the graphing part (1a):