Math Problem Statement

Describe how you can graph an exponential function with the help of doubling time or​ half-life. Choose the correct answer below.

A.

Start at the point

left parenthesis 0 comma Upper Q 0 right parenthesis0,Q0​,

where

Upper Q 0Q0

is the initial value of the function. For an exponentially growing​ quantity, the value Q of the function is

.5Upper Q 0Q0

after one doubling​ time,

.25Upper Q 0Q0

after two doubling​ times,

.125Upper Q 0Q0

after three doubling​ times, and so on. Fit a curve between these points.

B.

Start at the point

left parenthesis 0 comma Upper Q 0 right parenthesis0,Q0​,

where

Upper Q 0Q0

is the initial value of the function. For an exponentially growing​ quantity, the value Q of the function is

2Upper Q 0Q0

after one doubling​ time,

4Upper Q 0Q0

after two doubling​ times,

8Upper Q 0Q0

after three doubling​ times, and so on. Fit a curve between these points.

C.

Start at the point

left parenthesis 0 comma Upper Q 0 right parenthesis0,Q0​,

where

Upper Q 0Q0

is the initial value of the function. For an exponentially growing​ quantity, the value Q of the function is

2Upper Q 0Q0

after one doubling​ time,

4Upper Q 0Q0

after two doubling​ times,

6Upper Q 0Q0

after three doubling​ times, and so on. Fit a curve between these points.

Solution

The correct answer is:

B.

Explanation:
To graph an exponential function using doubling time, follow these steps:

  1. Start at the point (0,Q0)(0, Q_0), where Q0Q_0 is the initial value of the function.
  2. For an exponentially growing quantity:
    • The value of QQ doubles to 2Q02Q_0 after one doubling time.
    • The value of QQ doubles again to 4Q04Q_0 after two doubling times.
    • The value of QQ continues doubling to 8Q08Q_0 after three doubling times, and so on.
  3. Fit a smooth exponential curve through these points.

This process highlights the nature of exponential growth, where the quantity increases multiplicatively based on the doubling time.

Let me know if you would like further clarifications. Here are 5 related questions:

  1. How can you use the concept of half-life to graph a decaying exponential function?
  2. What is the general formula for exponential growth, and how does it relate to doubling time?
  3. How does exponential growth differ from linear growth when graphed?
  4. Can you derive doubling time from the exponential growth rate? If so, how?
  5. What are some real-world examples where exponential growth is observed?

Tip: When graphing exponential functions, ensure your axes are scaled appropriately to capture rapid changes in values effectively.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Doubling Time
Half-Life

Formulas

Q = Q0 * 2^t/d (for exponential growth)
Q = Q0 * (1/2)^t/h (for exponential decay)

Theorems

Properties of Exponential Growth and Decay

Suitable Grade Level

Grades 9-12