Question

In the equation \(y=2 x^{\mathrm{n}}\), in which \(x\) is an integer greater than 1, what is a possible value of \(n\) that will ensure that the expression has exponential growth? (A) 0 (B) 1 (C) 4 (D) Not sufficient information

Short answer

C: 4, because \(x^4\) represents exponential growth.

Step-by-step solution

Understanding Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value. In mathematical terms, this often means that as one variable increases, the output gets magnified more than linearly.

Analyzing Variable Components

In the equation \(y = 2x^n\), the growth of \(y\) depends on the values of \(x\) and \(n\). Given that \(x\) is greater than 1, \(n\) must be analyzed to see its impact on the growth of \(y\).

Evaluating Possible Values of \(n\)

To ensure exponential growth with \(y = 2x^n\), \(n\) must be such that \(x^n\) grows exponentially. For exponential growth, \(n\) should be greater than 1, turning the equation into a power function with an exponent larger than 1. This makes \(x^n\) grow rapidly as \(x\) increases.

Verifying Each Option

(A) If \(n = 0\), \(x^0 = 1\), and the equation becomes \(y = 2\), a constant, not exponential growth.(B) If \(n = 1\), the equation becomes linear \(y = 2x\), not exponential growth.(C) If \(n = 4\), \(y = 2x^4\), which results in exponential growth since \(x\) raised to a power greater than 1 grows much faster.(D) Informally assess whether additional information is necessary—the previous case (C) shows sufficiency.

Key concepts

Power Function

A power function refers to mathematical expressions of the form \(y = ax^n\), where \(a\) is a constant, \(x\) is a variable, and \(n\) is the exponent. These functions become particularly intriguing when the exponent, \(n\), is an integer. Each variation of \(n\) reveals different behaviors in the function’s growth pattern.

When the exponent is positive and greater than 1, like \(n = 4\) in our equation, the function \(y = 2x^4\) demonstrates growing behavior that isn’t just linear. It grows more rapidly as \(x\) increases. This contrasts with linear functions where \(n = 1\), which simply scale by a constant factor.

The significance of a power function in mathematics is its ability to model relationships where growth or change doesn’t occur uniformly. Thus, power functions are crucial in observing natural phenomenons like the spread of populations where higher powers of \(x\) result in larger jumps in output.

Integer Exponent

The term integer exponent refers to the power \(n\) that \(x\) is raised to in an equation like \(y = 2x^n\). Choosing an appropriate integer exponent can alter the behavior of a mathematical function dramatically.

If \(n\) is zero or one, as in options (A) and (B), the equation simplifies considerably. Specifically:

• \(n = 0\): Results in \(y = 2x^0 = 2\), a constant value, indicating no growth regardless of \(x\).
• \(n = 1\): Transforms into a linear equation, \(y = 2x\), which rises at a steady rate.

However, when an integer exponent is greater than one, the growth not only becomes faster but also non-linear.

When \(n\) is four, as in option (C), each unit increase in \(x\) leads to a much larger increase in \(y\), due to higher powers of \(x\) contributing more significantly to the value of \(y\). Understanding the implications of such an integer exponent provides insight into predicting how outputs increase as inputs grow.

Rapid Increase

Rapid increase in the context of power functions occurs when the value of a function rises steeply as \(x\) increases. In our equation \(y = 2x^n\) with \(n = 4\), the expression \(x^4\) signifies how quickly \(y\) escalates as \(x\) becomes larger. This rapid increase is typical of exponential growth scenarios.

Here's why such a situation is valuable:

• It models real-world phenomena where things get big quickly, like population growth or the spread of a virus.


• It explains why outputs in certain systems might seem to explode over time, giving insights into potential future trends.

As values like \(n\) exceed one, the function \(x^n\) leverages high exponents to amplify even small increases in \(x\) into significant leaps in \(y\). This non-linear correlation is key to understanding exponential growth.

Visualize it this way: if you plot \(y = 2x^4\), you’ll observe that for every tiny increment in \(x\), \(y\) shoots up much faster than when \(n\) is 0 or 1.