Q: Using the quadratic model $$U(x) = 75x^2$ + 35x + 100$$, estimate the number of mobile-only users in 2016 ($$x = 4).$$

$$U(4) = 75(4)^2$ + 35(4) + 100 = 75 \times 16 + 140 + 100 = 1200 + 140 + 100 = 1440$$ million

Q: Suppose the quadratic model $$U(x) = 75x^2$ + 35x + 100 is $$used to predict future users. What is the vertex of this parabola, and what does it represent in this context?

The vertex is at $$x = -\frac{b}{2a} = -\frac{35}{2 \times 75} = -0.233$$, representing a minimum, which does not make sense in this context.

Question 1

Which of the following best describes the general trend in Facebook's mobile-only users from 2012 to 2015, based on the data provided?

Accepted Answer

The number of users increases at an increasing rate.

Question 2

Would a linear function be a good model for the data on Facebook's mobile-only users from 2012 to 2015? Why or why not?

Accepted Answer

No, because the increase in users accelerates each year, suggesting a nonlinear model.

Question 3

What type of function, other than linear, might best model the growth in Facebook's mobile-only users from 2012 to 2015?

Accepted Answer

A quadratic function, because the rate of increase itself is increasing.

Question 4

If a quadratic function is used to model the data for Facebook's mobile-only users, will the parabola open upward or downward? Explain.

Accepted Answer

Upward, because the number of users is increasing at an increasing rate.

Question 5

Given the years 2012, 2013, 2014, and 2015 as $$x = 0, 1, 2, 3$$ respectively, and the corresponding users in millions as $$y = 100, 260, 450, 725$$, which quadratic function $$U(x) = ax^2$ + bx + c$$ fits the data?

Accepted Answer

$$U(x) = 75x^2$ + 35x + 100$$

Question 6

Using the quadratic model $$U(x) = 75x^2$ + 35x + 100$$, estimate the number of mobile-only users in 2016 ($$x = 4).$$

Accepted Answer

$$U(4) = 75(4)^2$ + 35(4) + 100 = 75 	imes 16 + 140 + 100 = 1200 + 140 + 100 = 1440$$ million

Question 7

When you use the quadratic model to estimate the number of mobile-only users in 2016, are you using interpolation or extrapolation?

Accepted Answer

Extrapolation, because 2016 is outside the range of the given data.

Question 8

Suppose the quadratic model $$U(x) = 75x^2$ + 35x + 100 is $$used to predict future users. What is the vertex of this parabola, and what does it represent in this context?

Accepted Answer

The vertex is at $$x = -\frac{b}{2a} = -\frac{35}{2 	imes 75} = -0.233$$, representing a minimum, which does not make sense in this context.

Question 9

If the trend in the data continues, which of the following best describes the long-term behavior of the quadratic model for Facebook's mobile-only users?

Accepted Answer

The number of users will increase without bound as time increases.

Question 10

Which of the following is a limitation of using a quadratic model to predict the number of Facebook mobile-only users far into the future?

Accepted Answer

Quadratic models can overestimate growth because they do not account for real-world limits such as market saturation.

Quadratic Functions and Modeling: Facebook Mobile Users Example

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