Identifying The Parent Function Of A Doubling Graph An Exploration Of Exponential Growth

Jul 12, 2025 by ADMIN 89 views

Understanding Exponential Growth The Parent Function of a Doubling Graph

When analyzing functions, a crucial aspect is understanding how the function's value changes with respect to its input. In this article, we will explore a specific scenario where the value of a graphed function doubles for each increase of 1 in the value of x. This behavior is characteristic of exponential functions, and we aim to identify the parent function that exhibits this doubling property. The question we will address is: If the value of a graphed function doubles for each increase of 1 in the value of x, which of the following could be the parent function of the graphed function?

Identifying the Doubling Behavior

To effectively determine the parent function, it's essential to grasp the concept of parent functions and exponential growth. A parent function is the simplest form of a function family, serving as a blueprint for other functions in that family. Transformations, such as stretches, compressions, and shifts, can be applied to the parent function to create more complex functions. In the context of our problem, we are looking for a parent function that, when graphed, demonstrates a doubling of its y-value for every unit increase in its x-value. This doubling behavior is a hallmark of exponential functions with a base of 2.

Let's consider the given condition: the function's value doubles for each increase of 1 in x. This implies that if f(x) is the function's value at x, then f(x+1) = 2f(x). This relationship is the core of exponential growth. Exponential functions have the general form y = ab^x, where a is the initial value, b is the growth factor, and x is the independent variable. In our case, since the function doubles, the growth factor b is 2. Therefore, we are looking for a function of the form y = a2^x*.

Analyzing the Given Options

We are presented with several options for the parent function:

y = 2^x
y = x^2
y = |2x|
y = √x

To determine which of these functions could be the parent function of the graphed function, we need to analyze each option and assess whether it exhibits the doubling behavior described in the problem.

1. y = 2^x

This function is an exponential function with a base of 2. Let's examine its behavior. When x increases by 1, the function becomes y = 2^(x+1). We can rewrite this as y = 2^x * 2^1 = 2 * 2^x. This clearly shows that the value of the function doubles for each increase of 1 in x. Therefore, y = 2^x satisfies the given condition and is a strong candidate for the parent function.

To further illustrate this, let's consider a few points on the graph of y = 2^x:

When x = 0, y = 2^0 = 1
When x = 1, y = 2^1 = 2
When x = 2, y = 2^2 = 4
When x = 3, y = 2^3 = 8

As we can see, the y-values double as x increases by 1, confirming the exponential growth behavior.

2. y = x^2

This function is a quadratic function. Let's analyze its behavior. When x increases by 1, the function becomes y = (x+1)^2. Expanding this, we get y = x^2 + 2x + 1. This is not a doubling of the original function y = x^2. The increase in the y-value is not a constant factor but depends on the value of x. Therefore, y = x^2 does not exhibit the doubling behavior and cannot be the parent function.

For example, let's consider a few points:

When x = 1, y = 1^2 = 1
When x = 2, y = 2^2 = 4
When x = 3, y = 3^2 = 9
When x = 4, y = 4^2 = 16

The y-values do not double consistently as x increases by 1. The increase is more significant for larger values of x.

3. y = |2x|

This function is an absolute value function. Let's analyze its behavior. The absolute value function y = |2x| outputs the magnitude of 2x, which means it reflects any negative values across the x-axis. When x increases by 1, the function becomes y = |2(x+1)| = |2x + 2|. This is not a doubling of the original function y = |2x|. The behavior of the absolute value function is linear in each segment (for positive and negative x), but it does not exhibit exponential growth.

Consider the following points:

When x = 1, y = |2(1)| = 2
When x = 2, y = |2(2)| = 4
When x = 3, y = |2(3)| = 6
When x = 4, y = |2(4)| = 8

Here, the y-values increase by 2 for each increase of 1 in x, indicating linear growth, not exponential doubling.

4. y = √x

This function is a square root function. Let's analyze its behavior. When x increases by 1, the function becomes y = √(x+1). This is not a doubling of the original function y = √x. The square root function exhibits a growth rate that decreases as x increases. It does not demonstrate the exponential doubling behavior described in the problem.

Let's look at some points:

When x = 1, y = √1 = 1
When x = 2, y = √2 ≈ 1.414
When x = 3, y = √3 ≈ 1.732
When x = 4, y = √4 = 2

The y-values do not double as x increases by 1. The increase in y becomes smaller as x increases.

Conclusion

Based on our analysis, the only function that exhibits the doubling behavior described in the problem is y = 2^x. For each increase of 1 in x, the value of the function doubles, which is a characteristic of exponential functions with a base of 2. Therefore, the parent function of the graphed function is y = 2^x.

In summary, understanding the properties of different types of functions, such as exponential, quadratic, absolute value, and square root functions, is crucial for identifying their parent functions and predicting their behavior. In this case, the exponential function y = 2^x perfectly matches the given condition of doubling its value for each unit increase in x.