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Chapter 5: Problem 9

One of the two tables below shows data that can be modeled by a linear function, and the other shows data that can be modeled by a quadratic function. Identify which table shows the linear data and which table shows the quadratic data, and find a formula for each model. $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 10 & 17 & 26 & 37 & 50 \\ \hline \end{array}\\\ &\text { Table A } \end{aligned} $$ $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline g(x) & 10 & 17 & 24 & 31 & 38 \\ \hline \end{array}\\\ &\text { Table B } \end{aligned} $$

Short Answer

Expert verified
Table A is quadratic with formula \( f(x) = x^2 + 6x + 10 \) and Table B is linear with formula \( g(x) = 7x + 10 \).

Step by step solution

01

Determine the Data Type for Table A

To determine if Table A shows linear or quadratic data, calculate the first differences of the function values, which are differences between consecutive values: \[ 17 - 10 = 7, \ 26 - 17 = 9, \ 37 - 26 = 11, \ 50 - 37 = 13. \] The first differences are not constant, indicating that Table A might not be linear. Let's check the second differences. Calculate the second differences: \[ 9 - 7 = 2, \ 11 - 9 = 2, \ 13 - 11 = 2. \] The second differences are constant, indicating that Table A shows data that can be modeled by a quadratic function.
02

Determine the Data Type for Table B

For Table B, calculate the first differences of the function values: \[ 17 - 10 = 7, \ 24 - 17 = 7, \ 31 - 24 = 7, \ 38 - 31 = 7. \] The first differences are constant, indicating that Table B shows data that can be modeled by a linear function.
03

Find the Quadratic Formula for Table A

Since Table A is quadratic, assume the formula is of the form \( f(x) = ax^2 + bx + c \). Use the data points to set up equations: \( f(0) = c = 10 \), \( f(1) = a(1)^2 + b(1) + 10 = 17 \), and \( f(2) = a(2)^2 + b(2) + 10 = 26 \). Solve these equations: \[ a + b + 10 = 17 \rightarrow a + b = 7, \] \[ 4a + 2b + 10 = 26 \rightarrow 4a + 2b = 16. \] Solve for \( a \) and \( b \): \[ 4a + 2b = 16 \] \[ 2a + b = 8 \] Subtract \( a + b = 7 \) from \( 2a + b = 8 \): \[ a = 1.0 \] Substitute back to find \( b \): \[ b = 6.0 \] So, \( f(x) = x^2 + 6x + 10 \).
04

Find the Linear Formula for Table B

Since Table B is linear, assume the formula is of the form \( g(x) = mx + c \). Use a given point and the slope, \( m = 7 \), from the constant first differences. Use \( g(0) = 10 \): \[ c = 10. \] Therefore, \( g(x) = 7x + 10 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
Linear functions are a type of function where the graph forms a straight line. The general form of a linear equation is \( g(x) = mx + c \), where \( m \) represents the slope, and \( c \) is the y-intercept.
In the context of Table B, we identified it as representing a linear function due to the constant first differences of the values, \( 7 \).
  • The slope \( m \) indicates how steep the line is. A higher value means a steeper incline.
  • The y-intercept \( c \) tells us where the line crosses the y-axis. For Table B, \( g(x) = 7x + 10 \), meaning the y-intercept is 10.
Understanding linear functions is crucial because they are fundamental to understanding more complex functions and represent real-world relationships like speed or cost.
Quadratic Functions
Quadratic functions are polynomial functions of the second degree, generally expressed in the form \( f(x) = ax^2 + bx + c \). Their graphs depict a U-shaped curve known as a parabola.
In Table A, we determined the data modeled a quadratic function due to the constant second differences, \( 2 \). This consistency signifies that the relationship between the values changes at a constant rate, different from linear functions' steady rate.
  • The coefficient \( a \) affects the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards. For Table A, \( f(x) = x^2 + 6x + 10 \).
  • The vertex of the parabola provides the maximum or minimum point, essential for identifying peaks in quantity or cost in real situations.
Quadratic functions are widely applicable in physics for projectile motion and economics for optimizing resources.
Difference Method
The Difference Method is a helpful tool for distinguishing between linear and quadratic functions. It involves analyzing the differences between successive data points.
  • For linear functions, the first differences (i.e., the differences between consecutive y-values) are constant.
  • For quadratic functions, the second differences (i.e., differences between first differences) are constant when the first differences are not.
By applying this method, it becomes easier to identify the underlying nature of functions that model real dynamics.
In our exercise, the Difference Method helped discern that Table A followed a quadratic pattern and Table B a linear one, facilitating the correct formulation and application of these mathematical models.
Algebraic Equations
Algebraic equations form the backbone of mathematical modeling, allowing us to build functions that explain or predict behaviors and phenomena.
In the context of modeling with functions, we solve algebraic equations to find expressions like \( ax^2 + bx + c \) for quadratic models or \( mx + c \) for linear models.
  • Formulating these equations requires careful selection of constants based on data or given points. Typically, equations are solved for constants using initial conditions or given values.
  • While linear equations are straightforward to set up and solve, quadratic equations require solving systems, possibly using substitution or elimination methods.
Gaining proficiency in forming and solving algebraic equations is vital, as it opens doors to understanding more complicated relationships and functions.

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Most popular questions from this chapter

One possible substitute for the logistic model of population growth is the Gompertz model, according to which Rate of growth \(=r N \ln \left(\frac{K}{N}\right)\). For simplicity in this problem we take \(r=1\), so this reduces to Rate of growth \(=N \ln \left(\frac{K}{N}\right)\) a. Let \(K=10\), and make a graph of the rate of growth versus \(N\) for the Gompertz model. b. Use the graph you obtained in part a to determine for what value of \(N\) the growth rate reaches its maximum. This is the optimum yield level under the Gompertz model with \(K=10\). c. Under the logistic model the optimum yield level is \(K / 2\). What do you think is the optimum yield level in terms of \(K\) under the Gompertz model? (Hint: Repeat the procedure in parts a and \(\mathrm{b}\) using different values of \(K\), such as \(K=1\) and \(K=100\). Try to find a pattern.)

Enzymes are proteins that act as catalysts converting one type of substance, the substrate, into another type. An example of an enzyme is invertase, an enzyme in your body, which converts sucrose into fructose and glucose. Enzymes can act very rapidly; under the right circumstances, a single molecule of an enzyme can convert millions of molecules of the substrate per minute. The Michaelis-Menten relation expresses the initial speed of the reaction as a rational function of the initial concentration of the substrate: $$ v=\frac{V s}{s+K_{m}}, $$ where \(v\) is the initial speed of the reaction (in moles per liter per second), \(s\) is the initial concentration of the substrate (in moles per liter), and \(V\) and \(K_{m}\) are constants that are important measures of the kinetic properties of the enzyme. \({ }^{74}\) For this exercise, graph the Michaelis-Menten relation giving \(v\) as a function of \(s\) for two different values of \(V\) and of \(K_{m}\). a. On the basis of your graphs, what is the horizontal asymptote of \(v\) ? b. On the basis of your graphs, what value of \(s\) makes \(v(s)=V / 2\) ? How is that value related to \(K_{m}\) ? c. In practice you don't know the values of \(V\) or \(K_{m}\). Instead, you take measurements and find the graph of \(v\) as a function \(s\). Then you use the graph to determine \(V\) and \(K_{m}\). If you have the graph, how will that enable you to determine \(V\) ? How will that enable you to determine \(K_{m}\) ?

Roughly \(90 \%\) of all stars are main-sequence stars. Exceptions include supergiants, giants, and dwarfs. For main-sequence stars (including the sun) there is an important relationship called the mass-luminosity relation between the relative luminosity \({ }^{25} L\) and the mass \(M\) in terms of solar masses. Relative masses and luminosities of several main-sequence stars are reported in the accompanying table. a. Find a power model for the data in this table. (Round the power and the coefficient to one decimal place.) The function you find is known to astronomers as the mass-luminosity relation. b. Kruger 60 is a main-sequence star that is about \(0.11\) solar mass. Use functional notation to express the relative luminosity of Kruger 60 , and then calculate that value. $$ \begin{array}{|l|c|c|} \hline \text { Star } & \begin{array}{c} \text { Solar mass } \\ M \end{array} & \begin{array}{c} \text { Luminosity } \\ L \end{array} \\ \hline \text { Spica } & 7.3 & 1050 \\ \hline \text { Vega } & 3.1 & 55 \\ \hline \text { Altair } & 1 & 1.1 \\ \hline \text { The Sun } & 1 & 1 \\ \hline 61 \text { Cygni A } & 0.17 & 0.002 \\ \hline \end{array} $$ c. Wolf 359 has a relative luminosity of about \(0.0001\). How massive is Wolf 359 ? d. If one star is 3 times as massive as another, how do their luminosities compare?

Ecologists have studied the relationship between the number \(S\) of species of a given taxonomic group within a given habitat (often an island) and the area \(A\) of the habitat. \({ }^{34}\) They have discovered a consistent relationship: Over similar habitats, \(S\) is approximately a power function of \(A\), and for islands the powers fall within the range \(0.2\) to \(0.4\). Table \(5.9\) gives, for some islands in the West Indies, the area in square miles and the number of species of amphibians and reptiles. a. Find a formula that models \(S\) as a power function of \(A\). b. Is the graph of \(S\) against \(A\) concave up or concave down? Explain in practical terms what your answer means. c. The species-area relation for the West Indies islands can be expressed as a rule of thumb: If one island is 10 times larger than another, then it will have _ times as many species. Use the homogeneity property of the power function you found in part a to fill in the blank in this rule of thumb. d. In general, if the species-area relation for a group of islands is given by a power function, the relation can be expressed as a rule of thumb: If one island is 10 times larger than another, then it will have _ times as many species. How would you fill in the blank? (Hint: The answer depends only on the power.) $$ \begin{array}{|l|r|c|} \hline \text { Island } & \text { Area } A & \begin{array}{c} \text { Number } S \\ \text { of species } \end{array} \\ \hline \text { Cuba } & 44,000 & 76 \\ \hline \text { Hispaniola } & 29,000 & 84 \\ \hline \text { Jamaica } & 4200 & 39 \\ \hline \text { Puerto Rico } & 3500 & 40 \\ \hline \text { Montserrat } & 40 & 9 \\ \hline \text { Saba } & 5 & 5 \\ \hline \end{array} $$

In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypes-that is, instances in which the two alleles carried at a particular site on an individual's chromosomes are both the same. For populations in which bloodrelated individuals mate, there is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. In general, the frequency of homozygous children from matings of blood-related parents is greater than that for children from unrelated parents. \(^{67}\) Measured over a large number of generations, the proportion of heterozygous genotypes-that is, nonhomozygous genotypes-changes by a constant factor \(\lambda_{1}\) from generation to generation. The factor \(\lambda_{1}\) is a number between 0 and 1 . If \(\lambda_{1}=0.75\), for example, then the proportion of heterozygous individuals in the population decreases by \(25 \%\) in each generation. In this case, after 10 generations the proportion of heterozygous individuals in the population decreases by \(94.37 \%\), since \(0.75^{10}=\) \(0.0563\), or \(5.63 \%\). In other words, \(94.37 \%\) of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For matings between siblings, \(\lambda_{1}\) can be determined as the largest value of \(\lambda\) for which $$ \lambda^{2}=\frac{1}{2} \lambda+\frac{1}{4}. $$ This equation comes from carefully accounting for the genotypes for the present generation (the \(\lambda^{2}\) term) in terms of those of the previous two generations (represented by \(\lambda\) for the parents' generation and by the constant term for the grandparents' generation). a. Find both solutions to the quadratic equation above and identify which is \(\lambda_{1}\). (Use a horizontal span from \(-1\) to 1 in this exercise and the following two exercises.) b. After 5 generations what proportion of the population will be homozygous? c. After 20 generations what proportion of the population will be homozygous?
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