Decide whether the word problem represents a linear or exponential function. Then, write the equation of the function. A library has 8000 books, and is adding 500 more books each y... Decide whether the word problem represents a linear or exponential function. Then, write the equation of the function. A library has 8000 books, and is adding 500 more books each year.
Understand the Problem
The question is asking to determine if the scenario described represents a linear or exponential function and then to form the equation of that function based on the provided information about the library's books.
Answer
Linear $y = 500x + 8000$
Answer for screen readers
The function is linear.
The equation is:
$$ y = 500x + 8000 $$
Steps to Solve
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Identify the Type of Function
The problem describes a library that has a fixed number of books (8000) and is adding a constant amount (500) of books each year. This indicates that the relationship is linear, as the same amount is being added each year. -
Formulate the Equation
In general, the equation of a linear function can be expressed as:
$$ y = mx + b $$
Where:
- $y$ is the total number of books after $x$ years.
- $m$ is the rate of change (the number of books added each year).
- $b$ is the initial value (the number of books initially present).
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Substituting Known Values into the Equation
From the information given:
- Initial number of books, $b = 8000$.
- Number of books added each year, $m = 500$.
Thus, the equation becomes:
$$ y = 500x + 8000 $$
The function is linear.
The equation is:
$$ y = 500x + 8000 $$
More Information
The library starts with 8000 books and increases its collection by 500 books each year. This consistent addition indicates a linear growth, as there's a steady increment over time.
Tips
- Confusing linear growth with exponential growth. Exponential functions involve growth at an increasing rate rather than a constant rate.
- Misinterpreting the initial condition or growth rate when forming the equation.
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