4. Applying Polynomial Functions to Real-World Situations
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Chapter 2
4.
Applying Polynomial Functions to Real-World Situations
Polynomial functions have a remarkable presence in numerous areas of study. The degree of the polynomial, a defining attribute of these functions, dictates their behavior and the nature of their graphs. This lesson emphasizes the various ways these functions get employed in tangible situations. Whether it's in physics to describe certain motions, in economics to predict trends, or in engineering for design purposes, the role of polynomial functions is undeniably significant. By grasping the concept of the degree and its influence on the function's characteristics, individuals can make more informed decisions and interpretations in their respective fields.
Lesson Settings & Tools
| | 10 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Applying Polynomial Functions to Real-World Situations
Slide of 10
Polynomials are used in a variety of fields to model diverse phenomena. This lesson aims to show how polynomials can model real-life situations.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Differences of Function Values
Consider the following table of values.
| x | f(x) |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 7 |
| 4 | 16 |
| 5 | 32 |
| 6 | 57 |
Looking at the x-values, it can be noted that the values are equally spaced. It is possible to find the first finite differences by subtracting two consecutive values. Consider the following example. f(2) - f(1) = 3 - 2 ⇓ f(2) - f(1) = 1
The first differences for each pair of consecutive terms can be found by doing the same process. Then, the second differences are found by subtracting consecutive first differences, third differences by subtracting consecutive second differences, and so on. Find every first, second, and third difference. Is there something that can be noted about the differences?
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Example
Writing a Function From a Graph
As part of an entrepreneur unit in school, Zosia is learning some interesting math. During her lesson, Zosia's teacher explained that modeling data is useful to understanding business. He explained that there are various methods to write a polynomial function that models data.
As an example, the teacher told the students about a certain company that has experienced ups and downs in earnings due to economic downturns.
- Before 1997, the company suffered losses due to the credits requested for the opening.
- In the years 1997, 2002, and 2007, the company had neither profit nor losses.
- From 1997 to 2002, the company made a profit.
- Between 2002 and 2007, the company suffered losses.
- From 2007 onwards, the company's profits have been increasing.
The teacher showed the students the following graph, where y represents the profit, in millions of dollars, and x represents the years before and after 2000. Here, negative values of y represent losses.
Then the teacher asked the students to write the cubic function graphed. Help Zosia write the expression for the cubic function.
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Hint
Notice that three of the points are x-intercepts.
Solution
The first thing to notice in the given graph is that three of the given points are zeros of the function. The Factor Theorem indicates that the zeros of a polynomial function are factors of the function. Since the function is cubic, the three zeros indicate the three factors.
(7,0), (-3,0), (2,0) [0.3em] ⇓ [0.3em]
f(x) = a(x-7)(x+3)(x-2)
In this expression, the factors are given by the zeros and a is a constant. To find the value of a, the fourth point is substituted into the expression.
f(x) = a(x-7)(x+3)(x-2)
SubstituteII
x= 0, f(x)= 7
7 = a( 0-7)( 0+3)( 0-2)
▼
Solve for a
IdPropAdd
Identity Property of Addition
7 = a(-7)(3)(-2)
Multiply
Multiply
7 = 42a
DivEqn
.LHS /42.=.RHS /42.
7/42=a
SimpQuot
Simplify quotient
1/6 = a
RearrangeEqn
Rearrange equation
a = 1/6
Now that the value of a has been found, it is possible to finish writing the function shown the graph, which represents the profits of the company. f(x) = 1/6(x-7)(x+3)(x-2)
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Discussion
Finite Differences and Their Properties
Sometimes it can be useful to examine the differences between function values when the input values are separated by a known quantity. These differences are known as finite differences.
Rule
Finite Difference
A finite difference is a mathematical expression that can be written in the following form.
f(x+c_1)-f(x+c_2)
In this expression, c_1 and c_2 are constants. There are three basic finite differences that are especially useful.
- Forward difference: f(x+h)-f(x)
- Backward difference: f(x)-f(x-h)
- Central difference: f(x+h)-f(x-h)
These differences are shown below.
Finite differences can be used to determine the degree of a polynomial function that models a data set. This is because finite differences have some really useful properties.
Rule
Properties of Finite Differences
The finite differences of polynomial functions satisfy the following two properties.
- For a polynomial function of degree n, the nth differences of the function values corresponding to equally spaced values of the independent variable are constant and different from zero.
- If the nth differences of equally spaced values are constant and different from zero, then the values can be represented by a polynomial function of degree n.
Proof
Informal JustificationTo justify these properties, the minor cases will be examined first.
n=1
Consider a linear function. f_1(x) = ax + b In this expression, a and b are constants different from zero. Suppose that the x-values are spaced by a constant h. The first differences between two arbitrary consecutive values of the function can be written as follows. f_1(x+h) - f_1(x) The expression for the linear function can be substituted here to see what happens.
Since both a and h are non-zero constants, their product is different from zero. Therefore, the first differences are always a non-zero constant, which confirms the property. It should be noted that, since the first differences are constant, all the second differences are zero.
n=2
Now consider a quadratic function. f_2(x) = ax^2 + bx + c In this expression, a, b, and c are non-zero constants. The values of x are separated by a constant h. A first difference can be obtained by subtracting two consecutive function values. f_2(x+h) - f_2(x) Again, it is possible to substitute the expression for f_2(x) to find an expression for the first difference.
f_2(x+h) - f_2(x)
SubstituteExpressions
Substitute expressions
a(x+h)^2 + b(x+h) + c - (ax^2 + bx + c )
▼
Simplify
Distr
Distribute -1
a(x+h)^2 + b(x+h) + c - ax^2 - bx - c
ExpandPosPerfectSquare
(a+b)^2=a^2+2ab+b^2
a(x^2 + 2xh + h^2) + b(x+h) + c - ax^2 - bx - c
Distr
Distribute a
ax^2 + 2axh + ah^2 + b(x+h) + c - ax^2 - bx - c
Distr
Distribute b
ax^2 + 2axh + ah^2 + bx+bh + c - ax^2 - bx - c
SubTerms
Subtract terms
2axh + ah^2 + bh
Looking at the linear term 2axh, it can be concluded that this difference is not constant. Now, try again with a second difference. Two consecutive first differences are needed to find a second difference. l f_2(x+h) - f_2(x) [0.25em] f_2(x+2h) - f_2(x+h) A second difference is obtained by subtracting the first expression from the second one.
f_2(x+2h) - f_2(x+h) - ( f_2(x+h) - f_2(x))
Distr
Distribute -1
f_2(x+2h) - f_2(x+h) - f_2(x+h) + f_2(x)
SubTerms
Subtract terms
f_2(x+2h) - 2f_2(x+h) + f_2(x)
SubstituteExpressions
Substitute expressions
a(x+2h)^2 + b(x+2h) + c -2(a(x+h)^2 + b(x+h) + c ) + ax^2 + bx + c
▼
Simplify
ExpandPosPerfectSquare
(a+b)^2=a^2+2ab+b^2
a(x^2+4xh+4h^2) + b(x+2h) + c -2(a(x^2+2xh+h^2) + b(x+h) + c ) + ax^2 + bx + c
Distr
Distribute a
ax^2+4axh+4ah^2 + b(x+2h) + c -2(ax^2+2axh+ah^2 + b(x+h) + c ) + ax^2 + bx + c
Distr
Distribute b
ax^2+4axh+4ah^2 + bx+2bh + c -2(ax^2+2axh+ah^2 + bx+bh + c ) + ax^2 + bx + c
Distr
Distribute -2
ax^2+4axh+4ah^2 + bx+2bh + c -2ax^2 -4axh - 2ah^2 - 2bx - 2bh - 2c + ax^2 + bx + c
SubTerms
Subtract terms
2ah^2
Since a and h are non-zero constants, the second difference is a constant different from zero, confirming the property again. Similar to the previous case, since the second differences are constant, the third differences are zero.
Conclusion
It can be concluded from the previous cases that, for a polynomial of degree n, its nth differences are the last constant non-zero differences. Therefore, the nth differences of a polynomial of degree n are constant and nonzero. Note that this is an informal justification and should not be taken as a proof.
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Example
Writing a Function From a Table of Values
Interested about Zosia's recent entrepreneur lessons, her uncle told her about his own experience. Some years ago, Zosia's uncle started a local business selling art supplies.
Her uncle showed Zosia how his sales went the first few years of business. At the end of the first year, he was $1000 behind because of a starting loan. Then he started seeing returns.
| Year | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Income | -$1000 | $1000 | $6000 | $14 000 | $25 000 |
Recalling her math lessons, Zosia decided to help her uncle by modeling the income with a polynomial function. Help Zosia do the following.
a Determine the degree of the polynomial function that better fits these values.
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b Let x be the year. Write the polynomial function.
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Hint
a Recall the Properties of Finite Differences.
b Use the corresponding regression function on a graphing calculator.
Solution
a To make the numbers smaller and easier to manage, the incomes can be divided by 1000. Doing so results in the following table.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | -1 | 1 | 6 | 14 | 25 |
By the Properties of Finite Differences, the constant differences can help determine the degree of the polynomial that best fits the data. The first differences are shown below.
The first differences are not constant. This means that the polynomial is not of degree 1 and that other differences need to be examined. Now it is time to look at the second differences.
Looking at the differences, it can be noted that the second differences are constant and non-zero. By the Properties of Finite Differences, the degree of the polynomial is 2.
b A graphing calculator can be used to find the polynomial function. To do so, first enter the given values into lists. Push STAT, choose Edit, and then enter the values in the first two columns. It should be noted that it is convenient to input the numbers divided by 1000.
To find a polynomial function of degree 2 to fit the data, the quadratic regression function of the calculator will be used. To view this function, press STAT, scroll to the right to view the CALC options, and choose the fifth option in the list, QuadReg.
This function is similar to a line of best fit function, but instead of fitting the data with a line, the quadratic regressions fits the data using a parabola. Using the information given, it is possible to write the function. y = 1.5x^2 - 2.5x Remember that at the beginning the numbers were divided by 1000. Therefore, to obtain the real income, each coefficient must be multiplied by 1000. y = 1500x^2 - 2500x It should be noted that the function can also be found by setting and solving a system of equations, where the coefficients are the variables. This method is explained later in the lesson.
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Pop Quiz
Finding the Degree of a Polynomial Function
Indicate the degree of the polynomial function that best fits the data in each table of values.
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Discussion
Polynomials That Perfectly Fit a Set of Points
When given a set of points, it is possible to determine a polynomial that perfectly fits these points. To do so, it is important to consider a principle called the (n+1) Point Principle.
Rule
The (n+1) Point Principle
Given a set of n + 1 points in the coordinate plane, where no pair of points lies on a vertical line, there is a unique polynomial of degree at most n that fits the points perfectly.
Proof
Informal JustificationTo justify this principle, the smaller cases will be examined to come to a conclusion.
n=0
Given a single point, it is possible to fit it with a unique constant function, or a zero degree polynomial.
When considering polynomials of greater degree, starting from degree 1, many different polynomials can include the point.
Therefore, the polynomials of greater degree that fit a single point are not unique.
n=1
Now consider that n=1. In this case, n+1=2 points are given. (x_1,y_1) (x_2,y_2) It is possible to write a system of equations by substituting these points into a linear function written in slope-intercept form. y_1=a x_1 +b & (I) y_2=a x_2 +b & (II) In these functions, a is the slope of the line and b its y-intercept. Solving this system of equations gives the values of a and b. The value of a can be zero if the points lie on a horizontal line, which means that the data can be fit with a constant function.
Similarly to the previous case, there are infinitely many different polynomials of greater degree that fit these two points, making the polynomials not unique.
Conclusion
In general, n+1 points generate a system of n+1 equations with n+1 variables, where each variable is a coefficient of a polynomial of degree n. y_1 = a_1 x_1^n + a_2 x_1^(n-1) + ⋯ + a_(n-1)x_1 + a_n y_2 = a_1 x_2^n + a_2 x_2^(n-1) + ⋯ + a_(n-1)x_2 + a_n ... y_n = a_1 x_n^n + a_2 x_n^(n-1) + ⋯ + a_(n-1)x_n + a_n A solution can result in a polynomial with a degree less than n if the leading coefficient a_1 is zero. It should be noted that this is an informal justification and should not be taken as a proof.
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Example
Writing a Polynomial That Fits a Set of Points
After seeing her enthusiasm about his experience, Zosia's uncle invited her out to visit and get some first hand experience with how a business works. Nervous about her first flight, Zosia watched a documentary about airplanes and their speeds at takeoff.
In the documentary, the horizontal distance traveled by the plane during the first few minutes after takeoff was analyzed.
- One minute after takeoff, the horizontal distance between the plane and the airport was 5 kilometers.
- Two minutes after takeoff, the plane was 9 kilometers from the airport.
- Three minutes after takeoff, the plane was 15 kilometers from the airport.
- Four minutes after takeoff, the plane was 24 kilometers from the airport.
Zosia thought she could model the distances by finding a polynomial that passes through the following four points. (1,5), (2,9) (3,15), (4,24) Here, the x-coordinates represent the minutes after takeoff and the y-coordinates represent the horizontal distance between the plane and the airport.
a Write a polynomial function for the plane's distance from the airport after takeoff. Round the answer to two decimal places if needed.
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b Use the polynomial found in Part A to estimate the distance between the plane and the airport 5 minutes after takeoff.
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Hint
a Use the (n+1) Point Principle.
b Substitute 5 for x in the function obtained in Part A.
Solution
a By the (n+1) Point Principle, when given four points, there may exist a polynomial of degree at most 3 that fits the points.
y = ax^3 + bx^2 + cx + d To find the values of the coefficients, the points are substituted into the equation above to generate a system of four equations. 5 = a(1)^3 + b(1)^2 + c(1) + d 9 = a(2)^3 + b(2)^2 + c(2) + d 15 = a(3)^3 + b(3)^2 + c(3) + d 24 = a(4)^3 + b(4)^2 + c(4) + d These equations can be rewritten by calculating the powers and rearranging the terms. 1a + 1b + 1c + 1d = 5 8a + 4b + 2c + 1d = 9 27a + 9b + 3c + 1d=15 64a + 16b + 4c + 1d = 24 Since the system is big, solving it by elimination or substitution can get messy. Because of this, it is preferred to use a graphing calculator to solve the system. To input a system of equations as a matrix, press 2ND, then x^(-1). Then, scroll to the right to reach the EDIT menu.
Selecting one of the matrices in the menu opens a window to modify its elements. To input a system of four equations with four variables, a 4*5 matrix will be needed. The last column stores the results at the right-hand side of each equation.
To solve the system of equations, press 2ND and x^(-1) once more. However, this time, select the MATH menu and scroll down until the rref(
option is selected.
Finally, input matrix [A] to the rref(
function. To do so, push 2ND and x^(-1) to bring up all available matrices. In the NAMES menu, choose matrix [A].
The solutions can be read vertically in the last column, using the remaining coefficients as reference points for the variables. [ cccc 1&0&0&0&0.166... 0&1&0&0&0 0&0&1&0&2.833... 0&0&0&1&2 ] ⇕ 1a + 0b + 0c + 0d = 0.166... 0a + 1b + 0c + 0d = 0 0a + 0b + 1c + 0d= 2.833... 0a + 0b + 0c + 1d = 2 Now that the coefficients have been calculated, it is possible to write the polynomial function. Remember to round each coefficient to two decimal places. y = 0.166... x^3 + 0x^2 + 2.833... x + 2 ⇓ y = 0.17x^3 + 2.83x + 2
b Substituting 5 for x in the equation found in Part A results in an estimate for the requested distance.
y = 0.17x^3 + 2.83x + 2
Substitute
x= 5
y = 0.17( 5)^3 + 2.83( 5) + 2
CalcPow
Calculate power
y = 0.17(125) + 2.83(5) + 2
Multiply
Multiply
y = 21.25 + 14.15 + 2
AddTerms
Add terms
y = 37.4
Therefore, the plane will be about 37.4 kilometers away from the airport 5 minutes after takeoff.
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Example
Predicting Carbon Dioxide Levels
Zosia is really into environmentalism. After watching the documentary about planes, she became increasingly worried about the atmospheric carbon dioxide levels on Earth and decided to do some research.
In her research, Zosia found data about the average levels, in parts per million, of carbon dioxide in the atmosphere per year. Some results are listed below. The years are counted from the year 1995, so 0 is the year 1995, 5 is 2000, and so on.
| Year | Carbon Dioxide Levels (ppm) |
|---|---|
| 0 | 360 |
| 5 | 369 |
| 10 | 379 |
| 15 | 390 |
| 20 | 401 |
| 25 | 414 |
Recalling her recent introduction to polynomial functions, Zosia decided to do the following.
a Use a calculator to find a polynomial of degree four that better fits the data in the table. Round every coefficient to four significant figures.
b Use the polynomial found in Part A to make a prediction for the year 2025. Is this prediction reliable? Round to four significant figures.
Answer
a C(t) = 6.667* 10^(-5)t^4 -0.003t^3 + 0.063t^2 + 1.538t +360.0
b 435.8 parts per million.
Hint
a Use the function
QuartRegon a graphing calculator.
b What does the value of R^2 indicate?
Solution
a The first thing that is needed is to input the data into the graphing calculator. To do so, press STAT, choose Edit, and enter the values in the first two columns.
To find a polynomial of degree four that fits the points, press STAT again and move to the CALC menu. Then, scroll down to select the function QuartReg.
Now that the coefficients are known, it is possible to write an function for the carbon levels. It is important to remember that the symbol E^(-5) is equivalent to multiplying the number by 10^(-5). C(t) = 6.667*10^(-5)t^4 -0.003t^3 + 0.063t^2 - 1.538t + 360.0
b To make the prediction for the carbon dioxide levels in 2025, this value has to be substituted into the function found in Part A. However, since the values are considered from 1995 as 0, the value that corresponds to 2025 is 2025-1995=30.
C(t) = 6.667*10^(-5)t^4 -0.003t^3+ 0.063t^2 + 1.538t +360.0
Substitute
t= 30
C( 30) = 6.667*10^(-5)( 30)^4 -0.003( 30)^3+ 0.063( 30)^2 + 1.538( 30) +360
C(30) = 435.8427
Therefore, the carbon dioxide levels in 2025 will be about 435.8 parts per million. Scrolling down on the last screen in Part A, it is possible to find the value of R^2. If R^2 does not appear, press 2ND, press , then select the DiagnosticOn
and press ENTER. Finally, reopen the window from Part A.
The closer the value of R^2 is to 1, the better the polynomial function fits the points. Since R^2 is very close to 1, the function fits the points really well. Therefore, it is possible that the prediction is reliable, but more points would clarify if that is the case.
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Closure
Using Polynomials to Fit the Given Data
The following table of values was presented earlier in the lesson.
| x | f(x) |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 7 |
| 4 | 16 |
| 5 | 32 |
| 6 | 57 |
Below are the first, second, and third differences of these values.
Note that the third differences are constant and different from zero. By the Properties of Finite Differences, the function values are represented by a cubic polynomial function. f(x) = ax^3 + bx^2 + cx + d Since many points are given, the best way to find the coefficients is to use the cubic regression function on a graphing calculator. The first thing to do is to input the data into the calculator. Push STAT, choose Edit, and then enter the values in the first two columns.
To find a polynomial of degree three that fits the points, press STAT again and move to the CALC menu. Then, scroll down to select the function CubicReg.
Now it is possible to write a polynomial function of degree three that fits the given data. y = 0.33x^3 - 0.5x^2 + 0.17x + 2
Since the value of R^2 is exactly 1, the polynomial function fits the points perfectly. This confirms the conclusion made considering the Properties of Finite Differences. In general, when encountering a data set, it can be useful to find a polynomial function to model the data, as it might lead to more accurate predictions or conclusions.
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Applying Polynomial Functions to Real-World Situations
Exercise 1.1
| x | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| f(x) | 8 | 76 | 256 | 596 | 1144 |
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Notice that the x-values in the table are equally spaced. To find the second differences, we begin by finding the first differences. We do this by subtracting each pair of consecutive function values.
Now that we know the first differences, we can proceed to finding the second differences. To do this, we subtract each pair of consecutive first differences.
As we can see, the second differences are 112, 160, and 208.
We will follow the same procedure as we did in Part A. However, notice that the given points are not in order. Let's first organize the points so that the x-values are in increasing order.
(1,-15), (3,169), (5,3001),
(7,16 689) (9,59 041), (11,161 305)
With the points organized, we can see that the x-values are equally spaced. Now let's begin by finding the first differences. As before, we will subtract each pair of consecutive y-values.
We are ready to find the second differences by subtracting the values we found for the first differences.
Repeating the same process one more time will give us the third differences.
The third differences for the given set of points are 8208, 17 808, and 31 248.
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Hint & Answer
S
Solution
| x | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| g(x) | -11 | -35 | -79 | -149 | -251 | -391 |
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According to the Properties of Finite Differences, if the nth differences of equally spaced values are constant and different from zero, then the values can be represented by a polynomial function of degree n. Let's begin by finding the first differences of the given function values.
As we can see, the first differences are not constant — the values are not equal. This means that a linear polynomial does not fit the data well. Let's continue and find the second differences.
Once more, we see that the second differences are not constant either. This means that a quadratic polynomial is not the best fit. Let's find the third differences.
This time, all the third differences are equal and different from zero — they are all equal to -6. This implies that the degree of polynomial function that best fits the data set is 3.
As in Part A, we will find the finite differences of the data set to determine the degree of the required polynomial. However, notice that the given points are not ordered. Therefore, let's organize them so that the x-values are in increasing order.
(2,-16),(3,-9),(4,0),(5,11),(6,24)
We can see that the x-values are equally spaced. Let's continue our process by finding the first differences. To do this, we subtract each pair of consecutive y-values.
Since the first differences are not constant, let's continue to find the second differences.
As we can see, the second differences are constant and different from zero. Therefore, the degree of the polynomial function that best fits the given data set is 2.
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Hint & Answer
S
Solution
Consider the following statement.
|
If the nth differences of equally spaced values are and zero, then the values can be represented by a polynomial function of degree n. |
Which of the following combinations completes the statement correctly?
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When we want to determine the degree of the polynomial function that best fits a set of values, we study the finite differences of y-values. In fact, if the nth differences of equally space values are constant and different from 0, the values can be represented by a polynomial function of degree n.
If the nth differences of equally spaced values are constant and different from zero, then the values can be represented by a polynomial function of degree n.
Consequently, the missing parts of the given statement are constant and different from.
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However, Davontay's teacher said that he made a mistake.
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Let's check Davontay's computations carefully to determine where he made a mistake. As we can see, he first wrote the points through which the cubic function passes. (3,0), (-2,0), (1,0), (4,72) A general cubic polynomial has the form ax^3+bx^2+cx+d. Since three of the given points are x-intercepts, we can use the factored form of a cubic polynomial, which is what Davontay did. ( 3,0), ( -2,0), ( 1,0) [0.75em] f(x) = a(x + 3)(x - 2)(x + 1) However, the factor corresponding to an x-intercept of the form (k,0) is x-k. In other words, to form the factor, the sign of the x-intercept is changed. This is where Davontay made his mistake — he forgot to change the signs. ( 3,0), ( -2,0), ( 1,0) [0.75em] f(x) = a(x + 3)(x - 2)(x + 1) * [0.75em] f(x) = a(x- 3)(x+ 2)(x- 1) ✓ Therefore, Davontay's mistake is that the sign on each factor is wrong.
To write the correct function rule, we can follow Davontay's steps but correct his mistake. Let's write the points through which the polynomial passes.
(3,0), (-2,0), (1,0), (4,72)
Since three of the given points are x-intercepts, we can use the factored form of a cubic polynomial.
f(x) = a(x-k_1)(x-k_2)(x-k_3)
Here, k_1, k_2, and k_3 are the x-intercepts. Let's substitute the corresponding values.
Our next step is to find the value of a. To do so, we can use the fact that the polynomial also passes through ( 4, 72), which implies that f( 4)= 72. Let's evaluate f(x) at x= 4.
We got that the value of a is 4. We are ready to write the required cubic polynomial. f(x) = 4(x-3)(x+2)(x-1)
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Applying Polynomial Functions to Real-World Situations
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