Question 1

Which type of function best models data that rises and then falls in a curved manner, such as the population trend shown in the first scatterplot (Image 8)?

Accepted Answer

A quadratic function $$f(x) = ax^2$ + bx + c$$, $$a \u003c 0$$

Question 2

Given the scatterplot of production workers' wages (Image 13), which type of function would be most appropriate to model the data?

Accepted Answer

A linear function $$f(x) = mx + b$$

Question 3

A researcher finds that the death rate for males as a function of hours of sleep per night fits the model $$y = 104.5x^2$ - 1501.5x + 601.6. $$What is the predicted death rate for males who sleep 8 hours per night?

Accepted Answer

$$y = 692$$

Question 4

Which of the following is the general form of a quadratic function that opens downward?

Accepted Answer

$$f(x) = ax^2$ + bx + c$$, $$a \u003c 0$$

Question 5

Suppose you are given the following data points: $$(1, 1.9)$$ and $$(4, 19.5). $$What is the slope $$m of $$the linear function $$P(x) = mx + b$$ that fits these points?

Accepted Answer

$$m = 5.87$$

Question 6

A scatterplot shows data that rises and falls more than once. Which type of function is most likely to fit this data?

Accepted Answer

A polynomial function of degree greater than 2

Question 7

Given the model $$P(x) = 5.87x - 3.97$$, what is the predicted percent increase in wages 14 years after 1996?

Accepted Answer

$$P(14) = 78.21\%$$

Question 8

Which of the following best describes the process of curve fitting in mathematical modeling?

Accepted Answer

Finding a function that approximates the general trend of the data

Question 9

If a scatterplot of CEO pay increases since 1990 (Image 12) shows a sharp rise and then a sharp fall, which type of function might best model this data?

Accepted Answer

A quadratic function

Question 10

Given the quadratic model $$y = 104.5x^2$ - 1501.5x + 601.6$$, what is the predicted death rate for males who sleep 2 hours per night?

Accepted Answer

$$y = 3431$$

Question 11

A data set shows a constant percent increase over time. Which function best models this situation?

Accepted Answer

An exponential growth function $$f(x) = a_0$ \cdot a^x$$, $$a \u003e 1$$

Question 12

Which of the following is NOT a characteristic of a linear model?

Accepted Answer

The graph has a turning point

Question 13

Given the data in the table for average number of live births per 1000 women by age (Image 15), which type of function would best fit the data?

Accepted Answer

A quadratic function

Question 14

If a data set fits the model $$f(x) = a_0$ \cdot a^x$$ with $$0 \u003c a \u003c 1$$, what does this indicate about the data?

Accepted Answer

The data is decreasing exponentially

Question 15

A scatterplot of driver fatalities by age (Image 11) shows a U-shaped curve. Which function best models this data?

Accepted Answer

A quadratic function $$f(x) = ax^2$ + bx + c$$, $$a \u003e 0$$

Question 16

Suppose you have three data points: $$(2, 34)$$, $$(5, 111.1)$$, and $$(10, 6.8). $$What type of function can you always fit exactly through these points?

Accepted Answer

A quadratic function

Question 17

Which of the following is a real-world example where curve fitting and mathematical modeling are essential?

Accepted Answer

All of the above

Question 18

If a data set fits a quadratic model $$f(x) = ax^2$ + bx + c$$ with $$a \u003e 0$$, what is the general shape of the graph?

Accepted Answer

A parabola opening upward

Question 19

Given the quadratic model $$y = 104.5x^2$ - 1501.5x + 601.6$$, what is the predicted death rate for males who sleep 10 hours per night?

Accepted Answer

$$y = 1451$$

Question 20

Which of the following statements about curve fitting is correct?

Accepted Answer

Curve fitting finds a function that best approximates the trend in the data

Mathematical Models and Curve Fitting: Foundations for Calculus

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