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Chapter 4: Problem 62

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

Short Answer

Expert verified
The scatter plot of a data set that suggests an exponential model usually starts near an asymptote (typically the x-axis) and then quickly increases or decreases. There is an evident growth or decay trend observed as the points move along the x-axis. The spacing between the points would tend to increase or decrease, showing an exponential rate of change.

Step by step solution

01

Understanding Exponential Function

An exponential function can be described as a mathematical function of the form f(x) = a * b ^ x, where a is not equal to zero, b is greater than zero but not equal to one, and x is any real number. The base b is constant and the exponent x varies.
02

Identifying Scatter Plot Shape for Exponential Function

The scatter plot shape that suggests the modeling of the data with an exponential function would likely show a pattern where the increase becomes rapidly larger as it moves along the x-axis. It starts slow and then increases or decreases steeply. It can either go upwards or downwards depending if the function is increasing or decreasing exponential.
03

Distinctive Characteristics of Scatter Plot for Exponential Function

In a scatter plot suggesting an exponential model, the points start near an asymptote (usually the x-axis) and then increase or decrease steeply. The points may be very close together, near the y-axis, and then become increasingly spaced out as they move away from the y-axis, which shows that the rate of change is increasing or decreasing exponentially.

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Most popular questions from this chapter

he formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 . a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(\left.2 A_{0}\right)\) is given by \(t=\frac{\ln 2}{k}\)

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$2^{x+1}=8$$

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=112.5 e^{0.012 y}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?
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