Question

Agronomists use radical functions to model and optimize corn production. One factor they analyse is how the amount of nitrogen fertilizer applied affects the crop yield. Suppose the function \(Y(n)=760 \sqrt{n}+2000\) is used to predict the yield, \(Y,\) in kilograms per hectare, of corn as a function of the amount, \(n,\) in kilograms per hectare, of nitrogen applied to the crop. a) Use the language of transformations to compare the graph of this function to the graph of \(y=\sqrt{n}\). b) Graph the function using transformations. c) Identify the domain and range. d) What do the shape of the graph, the domain, and the range tell you about this situation? Are the domain and range realistic in this context? Explain.

Short answer

The function is a vertically stretched and shifted version of \(y=\sqrt{n}\). Domain: \([0, \infty)\), Range: \([2000, \infty)\). This indicates yield increases as more nitrogen is applied.

Step-by-step solution

- Understand the Function

The function provided is \(Y(n)=760 \sqrt{n}+2000\). This means that yield \(Y\) of corn depends on the amount of nitrogen \(n\).

- Compare with \(y=\sqrt{n}\)

The function \(Y(n)=760 \sqrt{n} + 2000\) can be compared to \(y=\sqrt{n}\). The transformation is a vertical stretch by a factor of 760 and a vertical shift up by 2000 units.

- Graph Using Transformations

To graph \(Y(n)\), start with the basic square root function \(y=\sqrt{n}\). Stretch it vertically by multiplying the equation by 760, resulting in \(y=760 \sqrt{n}\). Then shift this graph up by 2000 units.

- Identify the Domain

The domain of \(Y(n)\) is the set of all possible values of \(n\). Since \(n\) represents the amount of nitrogen, it cannot be negative. Hence, the domain is \([0, \infty)\).

- Identify the Range

The range of \(Y(n)\) is the set of all possible values of \(Y\). Since the minimum value of \(760\sqrt{n}\) when \(n=0\) is 0, and adding 2000 gives 2000, the range is \([2000, \infty)\).

- Interpret the Graph, Domain, and Range

The shape of the graph shows that as the amount of nitrogen increases, the yield also increases, but at a decreasing rate. The domain and range being realistic means there is a minimum yield of 2000 kg/ha when no nitrogen is added, and the yield increases with added nitrogen.

Key concepts

Transformations

Transformation is a method used to adjust a function's graph. It includes shifts, stretches, and reflections.
In the given exercise, we start with the basic equation: \( y= \sqrt{n} \). This represents a square root function.
Now, let's perform the transformations step-by-step:

• Vertical Stretch: We multiply \( y= \sqrt{n} \) by 760 to get \( y= 760 \sqrt{n} \). This stretches the graph vertically.
• Vertical Shift: Then, we add 2000 to the equation, resulting in \( Y(n)= 760 \sqrt{n} + 2000 \). This shifts the graph upwards by 2000 units.

By performing these transformations, we get the final function. This new function models the crop yield of corn and shows how it changes with varying amounts of nitrogen.
Taking this approach helps in understanding how each parameter in the equation affects the overall graph.

Domain and Range

The domain of a function is the set of all possible input values. For our function \( Y(n)= 760 \sqrt{n} + 2000 \), the input is the amount of nitrogen (). Since nitrogen cannot be negative, the domain is \( [0, \infty) \).

The range is the set of all possible output values. By evaluating the function at the smallest domain value, we find:
\ n=0, \ Y(0) = 760 \sqrt{0} + 2000 = 2000. Thus, the minimum yield is 2000 kg/ha. As \ n \ increases, so does Y. Therefore, the range is \( [2000, \infty) \), meaning the yield can grow without limitation but will always be at least 2000.

• Domain: \( [0, \infty) \)
• Range: \( [2000, \infty) \)

These parameters enable agronomists to understand the limits within which the function operates. In practical terms, it shows the yield values with zero and increasing nitrogen supply.

Graph Interpretation

Interpreting the graph helps us understand the relationship between nitrogen fertilizer and corn yield. As nitrogen increases, the yield also increases but at a decreasing rate. This can be seen in the graph's shape, which starts steep and gradually flattens.

Here are some key points to note:

• The steep start indicates a rapid increase in yield with initial nitrogen application.
• As more nitrogen is added, the yield increase slows. This suggests diminishing returns for higher nitrogen levels.
• The graph’s vertical shift by 2000 shows an initial yield even with zero nitrogen. This could represent natural soil fertility.

The realistic domain and range reflect practical agricultural scenarios. Corn yield starts at 2000 kg/ha with no fertilizer, which is reasonable. The continued increase aligns with agronomists' observations of diminishing returns with more fertilizer.
Overall, the graph provides an easy visualization of how nitrogen affects yield, guiding decisions on optimal fertilizer levels.