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RF7 · Use polynomial functions of degree at most three to represent data and solve problems

Learn to use polynomial functions of degree at most three to represent data and solve problems through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Relations and Functions

Represent patterns and solve contextual problems with linear, quadratic, and cubic models

A set of measurements can show how one quantity changes as another quantity changes. A polynomial function can represent that pattern and help answer questions about values that were not measured. In this lesson, you will review function notation, connect polynomial degree to a model, and use models to solve problems. A model is a mathematical description of a situation. Its answers are useful only when they fit the context and the model’s assumptions.

What you will learn

  • Recognize polynomial functions of degree at most three.
  • Use data and a stated model to describe a situation.
  • Solve a contextual problem with a polynomial model and check whether the answer makes sense.

Review: variables, functions, and data

A variable is a letter that represents a number. In a function, an input value is paired with one output value. For example, if xx is the number of days after a plant is measured and h(x)h(x) is its height, then h(2)h(2) means the predicted height two days after the starting measurement.
A data point records an input and its measured output. A table or scatter plot can help show whether the data follow a pattern. The input and output also have units, such as days and centimetres. Keep those units in mind when interpreting a model.
  • The input is the quantity chosen or measured first; the output depends on it.
  • A model can estimate values between or beyond measured inputs, but estimates may be less reliable outside the observed range.

Polynomial models and degree

A polynomial function is made from terms that multiply a number by a variable raised to a whole-number power, then add or subtract those terms. The degree is the greatest power of the variable with a nonzero coefficient. In this expectation, use degree zero, one, two, or three.
A degree-one polynomial is linear and has a constant rate of change. A degree-two polynomial is quadratic and can model a curved pattern with a turning point. A degree-three polynomial is cubic and can have a more complex curved pattern. The shape is a clue, not proof that a particular model fits the situation.
To represent data, first inspect a table or scatter plot. Then choose a reasonable degree and find a model using given information or graphing technology. Polynomial regression is a technology method that finds a polynomial curve that fits a set of data as closely as possible. A regression model may not pass through every measured point. Report rounded coefficients as approximations, and use the model only where it makes sense for the situation.
f(x)=anxn+⋯+a1x+a0,0≤n≤3f(x)=a_nx^n+\cdots+a_1x+a_0,\quad 0\le n\le 3
  • The degree is the highest exponent, not the number of terms.
  • Use a model’s input and output units when explaining its predictions.
  • A close fit to data does not guarantee that the model is appropriate outside the measured range.

Use a model to answer a question

Once you have a model, substitute an input to predict an output. To find an input for a target output, set the model equal to that target and solve. A polynomial equation may have more than one solution, so check every answer against the context.
The domain is the set of input values that make sense for the problem. For example, time after an event cannot be negative. A model may also be unsuitable far beyond the inputs used to build it. State the relevant domain restriction and check the units of the answer.
  • Substitution answers an output question; solving an equation answers an input question.
  • Reject answers that violate the context, even if they satisfy the equation.

Practice and reasoning

When reading a model, identify what each variable represents before calculating. Decide whether the question asks for an output or an input. Then calculate, include units, and explain why the answer is reasonable.
For a data set, compare the model with the plotted measurements. Look for a consistent pattern rather than choosing a degree only because it produces a convenient equation. A model is an approximation of real data, so its predictions should be treated as estimates.
  • Use graphing technology to view the data and, when appropriate, fit a polynomial model.
  • Keep the chosen degree at three or less.

Worked example

Build a quadratic model from exact data

A seedling’s height is measured in centimetres, xx days after a starting measurement. The data are (0,2)(0,2), (1,5)(1,5), (2,10)(2,10), and (3,17)(3,17). The pattern is modelled by a quadratic function. Find the height predicted after five days, and find when the model predicts a height of 25 cm.
Seedling height model
Seedling height modelWindow of the quadratic model; the supplied data lie on this model.0123450510152025303540h(x)xyTime (days)Height (cm)● Supplied data

Window of the quadratic model; the supplied data lie on this model.

  1. Identify the pattern
    The first differences in the heights are 3, 5, and 7. The second differences are both 2, which is consistent with a quadratic pattern. The data fit the model shown.
    h(x)=x2+2x+2h(x)=x^2+2x+2
  2. Predict the height
    Substitute five days for the input. The output is a predicted height in centimetres.
    h(5)=52+2(5)+2=37h(5)=5^2+2(5)+2=37
  3. Solve for the target height
    Set the model equal to 25 and solve. Only a nonnegative day value is meaningful in this context.
    x2+2x+2=25  ⇒  x=−1±26x^2+2x+2=25\;\Rightarrow\;x=-1\pm2\sqrt{6}
Answer: The model predicts a height of 37 cm after five days. For a height of 25 cm, the solutions are approximately 3.90 days and −5.90 days. The negative time is outside the context, so the meaningful answer is about 3.90 days after the starting measurement.
Check: Substituting approximately 3.90 into the model gives an output close to 25. The other solution is negative and must be rejected for time after the starting measurement.

Worked example

Solve a problem with a cubic model

A community garden uses the model S(t)=−t3+6t2−9t+4S(t)=-t^3+6t^2-9t+4 for a seasonal score, where tt is time in weeks after the season begins and S(t)S(t) is a score in points. The model is intended for 0≤t≤40\le t\le4. At what times does the model predict a score of zero?
Seasonal score model
Seasonal score modelWindow of the stated cubic model on its intended time interval.01234-1012345S(t)xyTime (weeks)Score (points)

Window of the stated cubic model on its intended time interval.

  1. Set the score to zero
    The question asks when the output is zero, so set the model equal to zero. Factoring gives a product that is zero when either factor is zero.
    −t3+6t2−9t+4=−(t−1)2(t−4)-t^3+6t^2-9t+4=-(t-1)^2(t-4)
  2. Find and check the times
    The factors give times of one and four weeks. Both values are in the stated domain, so both are valid model predictions.
    (t−1)2(t−4)=0  ⇒  t=1,  4(t-1)^2(t-4)=0\;\Rightarrow\;t=1,\;4
Answer: The model predicts a score of zero at one week and four weeks.
Check: Both times lie in the model’s stated interval. Substitution gives a score of zero at each time.

Worked example

Interpret a regression model

A volunteer records the number of seedlings sprouted in a tray at several times. A quadratic regression using the supplied measurements returns the model N(d)=0.8d2+1.1d+2N(d)=0.8d^2+1.1d+2, where dd is days after planting and N(d)N(d) is the estimated number of seedlings. Use the model to estimate the count after five days. The measurements were taken from day zero to day four.
Seedling regression model
Seedling regression modelWindow of the given regression model; no measured data points are plotted.0123450481216202428N(d)xyTime (days)Estimated seedlings

Window of the given regression model; no measured data points are plotted.

  1. Substitute the requested input
    A regression model is an approximate fit to the recorded data. Substitute five days into the given model to make the requested estimate.
    N(5)=0.8(5)2+1.1(5)+2=27.5N(5)=0.8(5)^2+1.1(5)+2=27.5
  2. Interpret the estimate
    The model estimates a count, so a decimal output is not a literal count of seedlings. The requested day is also beyond the measured interval, which makes this an extrapolation: an estimate beyond the observed data.
    27.5≈2827.5\approx28
Answer: The model estimates about 28 seedlings after five days. This is an extrapolation beyond the day-zero-to-day-four measurements, so it should be treated cautiously.
Check: The calculation uses days as the input and produces an estimated seedling count. A count must be a whole number in the real situation, so the model’s decimal estimate is rounded for interpretation.

Common mistakes and how to avoid them

Calling a polynomial quadratic because it has three terms.
Correction: The degree is the largest exponent with a nonzero coefficient.
Keeping every algebraic solution, even when it gives negative time.
Correction: Check the stated context and domain before reporting a solution.
Treating a regression estimate as an exact measurement.
Correction: Regression gives an approximate model. State when a prediction is outside the measured input range.

Lesson summary

  • Polynomial models of degree at most three include linear, quadratic, and cubic functions.
  • Use data patterns or regression technology to select and find a model.
  • Substitute to predict an output; set the model equal to a target and solve to find an input.
  • Check units, context, and domain before interpreting a result.

Check your understanding

Question 1

What is the degree of g(x)=5−2x+7x3g(x)=5-2x+7x^3?
  1. 1
  2. 2
  3. 3
  4. 5
Show answer and explanation
3
The greatest exponent with a nonzero coefficient is three.

Question 2

A model is P(w)=w2+3w+2P(w)=w^2+3w+2, where ww is weeks after an event. Which solution to P(w)=0P(w)=0 is meaningful for this context?
  1. w=−1w=-1 only
  2. w=−2w=-2 only
  3. Both w=−1w=-1 and w=−2w=-2
  4. Neither solution
Show answer and explanation
Neither solution
The equation has solutions w=−1w=-1 and w=−2w=-2, but both are negative weeks after the event and are outside the context.

Question 3

A regression model was fitted to data measured from x=2x=2 to x=8x=8. What is a prediction at x=10x=10 called?
  1. Interpolation
  2. Extrapolation
  3. A measured value
  4. A degree
Show answer and explanation
Extrapolation
A prediction beyond the observed input range is extrapolation.

Key terms

Polynomial function
A function made by adding or subtracting terms whose variable powers are whole numbers.
Degree
The greatest exponent of the variable with a nonzero coefficient.
Polynomial regression
A technology method for finding a polynomial model that fits data approximately.
Domain
The set of input values that are allowed or make sense for a function or situation.
Extrapolation
Using a model to estimate an output beyond the range of observed inputs.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation RF7. It is a study resource, not an official curriculum publication.

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