Q: Given the linear model $$P(x) = 5.87x - 3.97$$ for the percent increase in wages $$x$$ years after 1996, what is the predicted percent increase in 2010?

$$P(14) = 5.87 \times 14 - 3.97 = 78.21\%$$

Q: Given the quadratic model $$y = 104.5x^2$ - 1501.5x + 601.6$$ for death rate $$y as a $$function of hours of sleep $$x$$, what is the predicted death rate for someone who sleeps 8 hours?

$$y = 104.5 \times 8^2$ - 1501.5 \times 8 + 601.6 = 692$$

Q: You are given the quadratic model $$y = -0.7197x^2$ + 38.2106x - 393.1272$$ for the average number of live births to women of age $$x. $$What is the predicted number of live births for women age 20?

$$y = -0.7197 \times 20^2$ + 38.2106 \times 20 - 393.1272 = 83.2$$

Question 1

Which of the following best describes a situation where a quadratic function with $$a \u003c 0 is $$the most appropriate model for a data set?

Accepted Answer

The data rise and then fall in a curved manner.

Question 2

Given the linear model $$P(x) = 5.87x - 3.97$$ for the percent increase in wages $$x$$ years after 1996, what is the predicted percent increase in 2010?

Accepted Answer

$$P(14) = 5.87 	imes 14 - 3.97 = 78.21\%$$

Question 3

A data set shows a rapid increase that slows down and then levels off. Which type of function is most likely to model this behavior?

Accepted Answer

Neither linear nor quadratic; likely a higher-degree polynomial or logistic function

Question 4

Suppose you have three data points: $$(2, 8)$$, $$(4, 18)$$, and $$(6, 32). $$Which type of function would you use to model the data if the rate of increase between points is not constant?

Accepted Answer

Quadratic function

Question 5

Given the quadratic model $$y = 104.5x^2$ - 1501.5x + 601.6$$ for death rate $$y as a $$function of hours of sleep $$x$$, what is the predicted death rate for someone who sleeps 8 hours?

Accepted Answer

$$y = 104.5 	imes 8^2$ - 1501.5 	imes 8 + 601.6 = 692$$

Question 6

Which of the following is the general form of a linear function used in curve fitting?

Accepted Answer

$$f(x) = mx + b$$

Question 7

A data set shows a steady decrease over time, but the rate of decrease slows as time increases. Which model best fits this data?

Accepted Answer

Exponential decay: $$f(x) = a_0$ \cdot a^x$$, $$0 \u003c a \u003c 1$$

Question 8

You are given the quadratic model $$y = -0.7197x^2$ + 38.2106x - 393.1272$$ for the average number of live births to women of age $$x. $$What is the predicted number of live births for women age 20?

Accepted Answer

$$y = -0.7197 	imes 20^2$ + 38.2106 	imes 20 - 393.1272 = 83.2$$

Question 9

Which of the following is NOT a characteristic of a quadratic function with $$a \u003e 0$$?

Accepted Answer

The graph opens downward.

Question 10

A biologist is modeling the population of bacteria in a petri dish. The population doubles every hour. Which function best models this growth?

Accepted Answer

$$f(x) = a_0$ \cdot 2^x$$

Question 11

If a data set rises and falls more than once, which type of function is most appropriate for modeling the data?

Accepted Answer

A polynomial function of degree greater than 2

Question 12

Given two data points $$(1, 1.9)$$ and $$(4, 19.5)$$, what is the slope $$m of $$the linear function passing through these points?

Accepted Answer

$$m = \frac{19.5 - 1.9}{4 - 1} = 5.87$$

Question 13

Which of the following is the general form of an exponential decay function?

Accepted Answer

$$f(x) = a_0$ \cdot a^x$$, $$0 \u003c a \u003c 1$$

Question 14

A researcher fits a quadratic model $$y = ax^2$ + bx + c to $$three data points. What is the minimum number of data points required to uniquely determine the coefficients $$a$$, $$b$$, and $$c$$?

Accepted Answer

Three

Question 15

A data set fits the model $$f(x) = |x|. $$Which of the following best describes the graph of this function?

Accepted Answer

A V-shaped graph symmetric about the y-axis

Question 16

Suppose a quadratic model $$y = ax^2$ + bx + c is $$used to fit data where $$a \u003c 0. $$What does this imply about the shape of the graph?

Accepted Answer

The parabola opens downward.

Question 17

A company’s revenue data over several years fits the model $$R(x) = 2x^2$ - 10x + 50$$, where $$x is $$years since 2000. What is the predicted revenue in 2005?

Accepted Answer

$$R(5) = 2 	imes 5^2$ - 10 	imes 5 + 50 = 50$$

Question 18

Which of the following is a key step in using curve fitting to make predictions from real-world data?

Accepted Answer

Selecting a mathematical model that best fits the pattern of the data

Question 19

If a data set fits a straight line but does not pass through the origin, which form should the linear model take?

Accepted Answer

$$f(x) = mx + b$$, where $$b 
eq 0$$

Mathematical Models and Curve Fitting in Calculus

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