The Equation and Its Components

Oct 13, 2025 by talynt.org 29 views

The given equation is $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ . This equation represents an exponential function with a negative base and a horizontal shift. In this article, we will delve into the details of this equation, its graphical representation, and the significance of the given data points.

The Equation and Its Components

The equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ consists of three main components:

Exponential term: $\left(\frac{1}{2}\right)^{x-2}$ represents an exponential function with a base of $\frac{1}{2}$ and an exponent of $x-2$ .
Negative sign: The negative sign in front of the exponential term indicates that the function is reflected about the x-axis.
Constant term: The constant term $-2$ is added to the exponential term, which shifts the function downward by 2 units.

Graphical Representation

The graph of the equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ is an exponential curve that decreases as $x$ increases. The graph has a horizontal asymptote at $y = -2$ , which means that as $x$ approaches infinity, the value of $y$ approaches $-2$ .

Given Data Points

The given data points are:

$x$	$y$
-1	-10
0	-6
1	-4
2	-3
3	$-\frac{5}{2} = -2.5$
4	$-\frac{9}{4} = -2.25$

These data points represent specific values of $x$ and the corresponding values of $y$ for the equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ .

Analyzing the Data Points

Let's analyze the given data points and their relationship with the equation.

For $x = -1$ , $y = -10$ : Substituting $x = -1$ into the equation, we get $y = -\left(\frac{1}{2}\right)^{-1-2} - 2 = -\left(\frac{1}{2}\right)^{-3} - 2 = -\frac{8}{1} - 2 = -10$ .
For $x = 0$ , $y = -6$ : Substituting $x = 0$ into the equation, we get $y = -\left(\frac{1}{2}\right)^{0-2} - 2 = -\left(\frac{1}{2}\right)^{-2} - 2 = -\frac{4}{1} - 2 = -6$ .
For $x = 1$ , $y = -4$ : Substituting $x = 1$ into the equation, we get $y = -\left(\frac{1}{2}\right)^{1-2} - 2 = -\left(\frac{1}{2}\right)^{-1} - 2 = -\frac{2}{1} - 2 = -4$ .
For $x = 2$ , $y = -3$ : Substituting $x = 2$ into the equation, we get $y = -\left(\frac{1}{2}\right)^{2-2} - 2 = -\left(\frac{1}{2}\right)^{0} - 2 = -1 - 2 = -3$ .
For $x = 3$ , $y = -2.5$ : Substituting $x = 3$ into the equation, we get $y = -\left(\frac{1}{2}\right)^{3-2} - 2 = -\left(\frac{1}{2}\right)^{1} - 2 = -\frac{1}{2} - 2 = -2.5$ .
For $x = 4$ , $y = -2.25$ : Substituting $x = 4$ into the equation, we get $y = -\left(\frac{1}{2}\right)^{4-2} - 2 = -\left(\frac{1}{2}\right)^{2} - 2 = -\frac{1}{4} - 2 = -2.25$ .

The given data points match the equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ , which confirms the validity of the equation.

Conclusion

$In conclusion, the equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ represents an exponential function with a negative base and a horizontal shift. The given data points match the equation, which confirms its validity. The graph of the equation is an exponential curve that decreases as $x$ increases, with a horizontal asymptote at $y = -2$.<br/> **Frequently Asked Questions (FAQs) About the Equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$**$

In this article, we will address some of the most frequently asked questions about the equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ .

Q: What is the base of the exponential term in the equation?

A: The base of the exponential term in the equation is $\frac{1}{2}$ .

Q: What is the exponent of the exponential term in the equation?

A: The exponent of the exponential term in the equation is $x-2$ .

Q: What is the horizontal asymptote of the graph of the equation?

A: The horizontal asymptote of the graph of the equation is $y = -2$ .

Q: What is the significance of the negative sign in front of the exponential term?

A: The negative sign in front of the exponential term indicates that the function is reflected about the x-axis.

Q: How does the equation change if the base of the exponential term is changed?

A: If the base of the exponential term is changed, the equation will change accordingly. For example, if the base is changed to $\frac{1}{3}$ , the equation becomes $y = -\left(\frac{1}{3}\right)^{x-2} - 2$ .

Q: How does the equation change if the exponent of the exponential term is changed?

A: If the exponent of the exponential term is changed, the equation will change accordingly. For example, if the exponent is changed to $x-1$ , the equation becomes $y = -\left(\frac{1}{2}\right)^{x-1} - 2$ .

Q: Can the equation be used to model real-world phenomena?

A: Yes, the equation can be used to model real-world phenomena that exhibit exponential decay. For example, the equation can be used to model the decay of a radioactive substance or the decrease in population of a species over time.

Q: How can the equation be used in applications?

A: The equation can be used in various applications such as:

Modeling population growth or decline: The equation can be used to model the growth or decline of a population over time.
Modeling radioactive decay: The equation can be used to model the decay of a radioactive substance over time.
Modeling chemical reactions: The equation can be used to model the rate of a chemical reaction over time.

Q: What are some common mistakes to avoid when working with the equation?

A: Some common mistakes to avoid when working with the equation include:

Incorrectly identifying the base or exponent: Make sure to correctly identify the base and exponent of the exponential term.
Incorrectly applying the negative sign: Make sure to correctly apply the negative sign in front of the exponential term.
Not considering the horizontal asymptote: Make sure to consider the horizontal asymptote of the graph of the equation.

Conclusion

In conclusion, the equation $y = -\left(\frac{1}{2}\right)^{x-2} - 2$ is a powerful tool for modeling real-world phenomena that exhibit exponential decay. By understanding the components of the equation and how to apply it, you can use it to model a wide range of phenomena and make predictions about the future.