Q: Which of the following best describes the end behavior of the cubic function $$y = x^3$ - 3x^2$ - 18x + 10$$?

As $$x \to \infty$$, $$y \to \infty$$; as $$x \to -\infty$$, $$y \to -\infty.$$

Question 1

Which of the following best describes a 'complete graph' as used in college algebra?

Accepted Answer

A graph that displays all relevant features of a function, including intercepts, asymptotes, and end behavior, within the domain of interest.

Question 2

Given the function $$y = x^3$ - 3x^2$ - 18x + 10$$, which window settings on a TI-84 calculator would best reveal the overall shape and key features of its graph?

Accepted Answer

Window: $$x_{min} = -10$$, $$x_{max} = 10$$, $$y_{min} = -25$$, $$y_{max} = 10$$

Question 3

Suppose the cost $$C of $$removing $$p	ext{\%} of $$pollution from drinking water is given by the model $$C = \frac{5350p}{100-p}. $$What is the domain of $$p$$ for which this model is valid?

Accepted Answer

$$0 \leq p \u003c 100$$

Question 4

Using the cost model $$C = \frac{5350p}{100-p}$$, what happens to the cost $$C as$$ $$p$$ approaches $$100$$?

Accepted Answer

The cost $$C$$ increases without bound (approaches infinity).

Question 5

If $$p = 90$$, what is the cost $$C to $$remove $$90\% of $$the pollution according to the model $$C = \frac{5350p}{100-p}$$?

Accepted Answer

$$C = 48,150$$ dollars

Question 6

Given the function $$y = x^3$ - 3x^2$ - 18x + 10$$, which of the following is a correct step to sketch its graph using a graphing calculator?

Accepted Answer

Enter the equation in the Y= editor and adjust the window to display key features such as intercepts and turning points.

Question 7

In the context of scatterplots, why is it important to align the data so that $$x = 0$$ corresponds to the starting year (e.g., 2015)?

Accepted Answer

It allows for easier interpretation of trends over time and simplifies regression analysis.

Question 8

If the number of U.S. adults with diabetes in 2015 is $$37,300 (in $$thousands), and in 2050 is projected to be $$100,000 (in $$thousands), what is the average annual increase over the $$35$$ years?

Accepted Answer

$$1,790 (in $$thousands) per year

Question 9

When creating a scatterplot of diabetes data on a TI-84 calculator, which list should contain the years since 2015?

Accepted Answer

List L1

Question 10

Which of the following is a limitation when using the model $$C = \frac{5350p}{100-p}$$ for pollution removal?

Accepted Answer

The model is undefined for $$p = 100$$ because the denominator becomes zero.

Question 11

If you use the TABLE feature on a TI-84 calculator to find $$C$$ when $$p = 90 in $$the model $$C = \frac{5350p}{100-p}$$, what steps should you follow?

Accepted Answer

Enter the equation in Y=, set the table to start at $$p = 90$$, and read the corresponding $$C$$ value.

Question 12

Which of the following best describes the end behavior of the cubic function $$y = x^3$ - 3x^2$ - 18x + 10$$?

Accepted Answer

As $$x 	o \infty$$, $$y 	o \infty$$; as $$x 	o -\infty$$, $$y 	o -\infty.$$

Question 13

If you want to graph $$y = x^3$ - 3x^2$ - 18x + 10$$ and see all intercepts, which window setting would be most appropriate?

Accepted Answer

$$x_{min} = -10$$, $$x_{max} = 10$$, $$y_{min} = -25$$, $$y_{max} = 25$$

Question 14

When plotting the diabetes data for years 2015, 2020, 2025, 2030, 2035, 2040, 2045, and 2050, what should the $$x-$$values be if you align the data so that $$x = 0$$ corresponds to 2015?

Accepted Answer

$$0, 5, 10, 15, 20, 25, 30, 35$$

Question 15

If you want to create a scatterplot of the diabetes data on a TI-84 calculator, which command should you use?

Accepted Answer

STAT PLOT

Question 16

Which of the following is a real-world implication of the cost model $$C = \frac{5350p}{100-p}$$ for pollution removal?

Accepted Answer

Removing nearly all pollution ($$p$$ close to $$100) $$becomes extremely expensive.

Question 17

If the diabetes data is entered in L1 and L2 on a TI-84 calculator, which list should contain the number of adults (in thousands)?

Accepted Answer

L2

Question 18

Which of the following statements about graphing functions is correct?

Accepted Answer

Adjusting the window settings is necessary to view all important features of a function.

Question 19

If you want to interpret the coordinates of the point where $$p = 90 in $$the cost model $$C = \frac{5350p}{100-p}$$, what do the $$x$$ and $$y$$ values represent?

Accepted Answer

$$x is $$the percentage of pollution removed, $$y is $$the cost in dollars.

Graphs of Functions, Graphing Calculator Techniques, and Data Modeling in College Algebra

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