Determine whether y = (x - 7)³ represents an exponential function.
Understand the Problem
The question is asking whether the given function, y = (x - 7)³, qualifies as an exponential function, which usually takes the form y = a * b^x, where a is a constant and b is a positive real number.
Answer
The function $y = (x - 7)³$ does not represent an exponential function.
Answer for screen readers
The function $y = (x - 7)³$ does not represent an exponential function.
Steps to Solve
- Identify the function form
The given function is $y = (x - 7)³$. An exponential function typically has the form $y = a \cdot b^x$, where $a$ is a constant and $b$ is a positive real number.
- Analyze the terms
In the function $y = (x - 7)³$, the exponent is applied to a linear expression $(x - 7)$, which indicates that the function is polynomial in nature, specifically a cubic function.
- Compare with exponential form
Since an exponential function requires the variable $x$ to be in the exponent (e.g., $b^x$), and in our function, $x$ is not in the exponent but rather in a polynomial term, this cannot be classified as an exponential function.
The function $y = (x - 7)³$ does not represent an exponential function.
More Information
Exponential functions grow at a rate proportional to their current value, while polynomial functions (like the given function) grow at a rate proportional to their degree. This fundamental difference classifies them into distinct categories.
Tips
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