3.04 Modeling with polynomial functions

Identify the important factors that need to be considered when answering this question.

Important factors affect the decision being made but they may not be explicitly stated in the problem. Our goal is to think of questions that need to be answered and identify these underlying aspects.

Some important factors are:

• What will the independent variable represent?

• What will the dependent variable represent?

• Will there be any restrictions on the variables?

• How long are average roller coasters?

• How tall are average roller coasters?

• How long do average roller coasters last?

• Can roller coasters begin and end at different heights?

There may be other factors that are important, but you may not think of them until later in the modeling cycle. This list will be continuously revised, and the model may need to change too. That is normal and expected while working through the modeling cycle.

List assumptions that need to be made.

Because the question is broad and there are many factors that could affect our decision, we need to make assumptions to narrow our focus.

A roller coaster's speed is variable throughout the ride. The ride moves slowly as it climbs a hill, and it moves quickly as it descends a hill. We do not have the skills or mechanics to calculate the speeds of the coaster, so we can assume that the plan for the roller coaster is focused more on its shape and design.

Similar to the previous part, there may be other assumptions that need to be made later in the modeling cycle. It is important to continue writing down each assumption that is made throughout the process.

State a recommended plan for solving the problem, including any research that needs to be done to further clarify the problem.

To propose a plan, we need to address each of the important factors we previously listed by conducting research to clarify the factor or by creating an action item to account for the factor.

A proposed plan for solving the problem may be as follows:

• To focus on the shape and design of the coaster, the independent variable should represent length. We still need to decide on the unit of measure, so we can research how other coasters are measured.

• The dependent variable will represent the height of the coaster. It will need the same unit of measure as the independent variable.

• Since the variables are lengths, the domain is restricted to positive values. We still need to decide if the x-axis represents ground level or the height at which the riders board the coaster.

• We need to research the average length and heights of roller coasters.

Ideally, this plan will give us the information we need to create a good model for a roller coaster.

Create a model for the path of the ball thrown by Cade.

We need to decide what the phrase "twice as far" means. To throw it twice as high, Cade would have had to throw the ball much harder than Jax. But because the boys had the same athletic skill, this doesn't seem reasonable.

An assumption we can make is that "twice as far" refers to horizontal distance. This means we will need to find the rightmost x-intercept of Jax's throw, then create a model where the rightmost x-intercept is twice the distance.

To determine the x-intercepts of Jax's throw, we can graph the function using technology.

Therefore, the function modeling Cade's throw would be C\left(x\right)=-7.98\left(x+0.103\right)\left(x-6.08\right). Now, we need to graph both functions on the same coordinate plane.

Analyze the model you created to determine if it is a good fit for the situation. If it is a good fit, explain why. If it is not a good fit, adapt the model to better fit the situation.

To analyze the model, we need to look at the key features in context. We have already looked at the intercepts, so we now need to determine if the other values make sense in context.

According to our model, Cade's ball does land twice as far as Jax's ball. However, Cade's ball also goes much higher than Jax's. That means that Cade would need to be much stronger than Jax in order to throw it that much higher. The information told us to assume the boys were the same size, so it would be highly unlikely that Cade was that much stronger than Jax.

We now need to make a new assumption and adjust the model. The new assumption is that Cade is not able to throw the ball much higher than Jax. The ball needs to reach a similar height as Jax's ball but travel twice as far horizontally.

Under this new assumption, Cade's throw can be modeled best by applying a horizontal stretch to the model of Jax's throw. Stretching the function for Jax's ball by a factor of 2 will make the ball reach the same height but travel twice as far. An algebraic model is:

Graphing J\left(x\right) and C\left(x\right) helps us compare the paths of both throws and confirm that Cade's ball travels twice the distance of Jax's, without going any higher.

Creating the model and analyzing the model are typically done at the same time in real-world situations. When creating a model, it is important that we analyze whether it make sense in the given context without being prompted. If it doesn't, we must go back and adjust it.

Adjusting the model or trying another route is common when modeling real-world situations as it can be difficult to get something right the first time around.

Explain how Cade was able to throw the ball twice as far.

In the previous part, we assumed that the boys were of similar strength since they were the same size. To get an idea of how Cade managed to throw the ball further than Jax, we can analyze the graph of both throws.

From the graph, we can see that the boys threw the ball from the same height and the ball reached the same height in both throws. The difference between the throws is the angle of elevation. Jax threw the ball straight into the air, so his throw has a steeper incline. The angle of elevation of Cade's ball is smaller, so the ball traveled further.

Other assumptions about the height could also make sense in this context. For example, someone may have assumed that Cade's ball would travel twice as far but only reach 75\% the height of Jax's ball. This assumption could also have created a valid model for the situation.

Create a model to represent the perimeter and area for each garden type. Define your variables and discuss any constraints.

To help us visualize this situation, a diagram would be a good model to start with. The diagram can help us create equation models which will then help us solve the question in part (b).

Because Kala wants to separate the fruit from the vegetables with a row of fencing, we need to include that part of the fence in our calculations. The perimeter will be the fence around the outside and the fence on the inside.

Let x represent the width of the rectangle which is the same length as the fence in the middle. Let y represent the length. Then, the equation for the perimeter is3x+2y=300 The area function is A\left(x,y\right)=xy

If we solve the perimeter equation for y, we get \begin{aligned}3x+2y&=300\\2y&=300-3x\\y&=150-1.5x\end{aligned} If we substitute 150-1.5x for y in the area formula, we can get a better understanding of the function.\begin{aligned}A\left(x\right)&=x\left(150-1.5x\right)\\&=150x-1.5x^2\end{aligned}

Because x represents a length, we know it must be a positive value. We can determine the upper limit of x by finding when either function is greater than zero. \begin{aligned}150-1.5x&>0\\150&>1.5x\\100&>x\end{aligned} This means 0<x<100.

For the circle, the perimeter can also represent the outside of the circle (the circumference) plus the fencing that goes down the middle.

Let r represent the radius of the circle. Then the perimeter equation is 2\pi r+2r=300. Since this is in one variable, we can solve for r.

The area of the circle can be represented by A=\pi r^2. Since we know the radius, we can substitute and solve this to find the size of the circle that is possible with the amount of fencing available.

With 300\text{ ft} of fencing, a circle with an area of 4\,120.964\text{ ft}^2 can be constructed.

Analyze your models to determine which type of garden Kala should make. Include the dimensions of the garden. Give mathematical and contextual arguments to defend your choice.

To answer this question, we will need to make an assumption about what key features Kala and her family might prefer. A rectangular shape is typical and may be easier to create, or may better fit a plot of land. A circle may waste space, or it may be difficult to plant seeds around the edge of the circle. But maybe Kala isn't concerned about the shape. Maybe she simply wants the shape that has the largest area.

In part (a), we found the area of a circular garden with the amount of fencing available is about 4\,121\text{ ft}^2. Graphing the area function of the rectangular garden can help us make decisions about selecting the shape Kala should choose for the garden.

Finding the vertex algebraically or looking at the graph, we find that the maximum area of a rectangular enclosure is 3\,750\text{ ft}^2. So, the area of the rectangular garden will always be less than the area of the circular garden.

Let's assume that Kala prefers gardening in rows. Although the circular garden has more space, the edges of the circle might end up wasting space which would make the area similar to the area of the rectangle since there is only a difference of a few hundred square feet.

With these factors in mind, Kala should build a rectangular garden. Assuming she wants to maximize the area, we can use the x-value of the vertex of the parabola to find the width of the garden. The width of the garden would be 50\text{ ft}, and the length of the garden would be:

Therefore, the dimensions of the rectangular garden would be 50\text{ ft} by 75\text{ ft}.

Our original diagram and equation models represented the situation accurately. But when we analyzed our models to choose a garden shape, we realized that a graph was also helpful to visualize the situation. Sometimes one type of model alone is not sufficient to find a solution. Multiple models were used to make sense of this problem, including diagrams, function equations, and graphs.

Explain which type of regression model you think best fits the data.

Plotting the data values will give insight about which type of function might best model the data. To plot the values, we need to define the variables. Let x represent the number of the month where March is month 1, April is month 2, etc.

This data could be represented by a few different polynomials, so it is important to defend our choice. Note that the curve does not need to pass through every point, but the model would be more accurate if it included as many points as possible.

The data begins increasing, then decreases around x=6. Therefore, a quadratic function appears to be a good function type for this data.

Another student may have argued that a cubic function would fit this best because the point at x=1 appears to be a turning point. Since there is also a turning point around x=6, this must be at least a cubic function.

Someone else may have said a quartic would be best because the data on the right side appears to decrease at a decreasing speed as it would just before a third turning point. Analyzing the data in this way helps us decide which type of model to start with when using technology to find a regression model.

Two previously collected data points were accidentally overlooked when the table was created. The added data points are: \begin{aligned}\text{Jan }1.57\\\text{Feb }1.14\end{aligned} Explain how this affects the type of function you would choose to fit the data.

We had previously defined the independent variable as the number of the month beginning from March. Now, the two months before March have been added, so we need to redefine x. Let x represent the number of the month where January is month 1, February is month 2, March is month 3, etc.

Now, let's re-plot the data values to gain insight on which type of function may be used to model the data.

The added data do affect the type of function we chose in part (a). It is clear that this data now has 2 turning points, so it now appears that a cubic function would fit the data best.

This answer will depend on what type of model was originally chosen in part (a). If a cubic function was already chosen in part (a), then these added data might not affect the decision.

Create a regression model that best fits all the data from the full calendar year. Justify the reasoning for your choice of model.

To create a regression model, we can use a technological tool. To use the tool, we need to enter the given data into a table, then find the polynomial regression of the desired degree.

One technology tool we can use to calculate the regression equation is the statistics calculator in GeoGebra.

1. Enter the x- and y-values in two separate columns, then highlight the data and select Two Variable Regression Analysis:

   

2. Choose Polynomial under the Regression Model drop down menu, then choose 3 under the next drop down menu to find the cubic of best fit:

   

The cubic curve of best fit is okay, but there are several points above and below the curve. We can use Geogebra to try to find a higher degree polynomial that fits the data better. Using the drop down menu on the right, we will choose 4 to examine the quartic curve of best fit.

This curve fits the data much better than the cubic because there are more points on or near the curve.

Therefore, a regression model that can be used to fit the data is f\left(x\right)=0.0027x^4-0.082x^3+0.7866x^2-2.4923x+3.4312.

For this data set, as the degree of the polynomial increases, the curve seems to more accurately model the given data. A 5th degree polynomial would be an even better match.

When finding the best fit for data, statisticians do not just estimate the curve of best fit based on a visual inspection of the graph. Instead, they make other mathematical calculations about their models to quantify how closely the data fits the regression. Statisticians and scientists would also consider features of the both the graph of a function and the context that might lead to choosing a particular model.

The city council wants a report about the rainfall over the course of the year. Explain to the council what information the model created in part (c) can provide. Include any limitations of the model.

The model can be used to fairly accurately evaluate the amount of rainfall that fell in the weeks between data collection points. We could calculate the rainfall at 4.25 months to find out how much rain there was the first week of April, for example.

However, the polynomial function model will not be useful to extend beyond December because a polynomial function will ultimately have end behavior that goes to positive or negative infinity. In other words, the model has domain restrictions from January to December of that particular year. The rainfall for the next year is more likely to repeat a similar pattern to the current year.

Because this data is likely to increase and decrease year after year, a different type of function may be a better fit for modeling multiple years of rainfall. We will learn about other functions that may end up being a better fit for this type of data later in Algebra 2.