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:
- Start at the point , where is the initial value of the function.
- For an exponentially growing quantity:
- The value of doubles to after one doubling time.
- The value of doubles again to after two doubling times.
- The value of continues doubling to after three doubling times, and so on.
- 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:
- How can you use the concept of half-life to graph a decaying exponential function?
- What is the general formula for exponential growth, and how does it relate to doubling time?
- How does exponential growth differ from linear growth when graphed?
- Can you derive doubling time from the exponential growth rate? If so, how?
- 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
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