Which Graph Represents The Function $y = 5 \cdot \left(\frac{1}{3}\right)^x$?

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Introduction

When it comes to graphing functions, understanding the behavior of the function is crucial in determining the shape of the graph. In this article, we will explore the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$. This function is an exponential function, and we will analyze its behavior to determine which graph represents it.

Understanding Exponential Functions

Exponential functions are a type of function that has the form $y = a \cdot b^x$, where $a$ and $b$ are constants. In this case, the function is $y = 5 \cdot \left(\frac{1}{3}\right)^x$. The base of the exponent is $\frac{1}{3}$, which is a fraction between 0 and 1. This means that the function will decrease as $x$ increases.

Graphing the Function

To graph the function, we need to understand its behavior. Since the base of the exponent is $\frac{1}{3}$, the function will decrease as $x$ increases. This means that the graph will be a decreasing curve.

Finding the y-Intercept

The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, we need to substitute $x = 0$ into the function. This gives us:

$$y = 5 \cdot \left(\frac{1}{3}\right)^0 = 5 \cdot 1 = 5$$

So, the y-intercept is at the point (0, 5).

Finding the x-Intercept

The x-intercept is the point where the graph intersects the x-axis. To find the x-intercept, we need to substitute $y = 0$ into the function. This gives us:

$$0 = 5 \cdot \left(\frac{1}{3}\right)^x$$

Dividing both sides by 5, we get:

$$0 = \left(\frac{1}{3}\right)^x$$

Taking the logarithm of both sides, we get:

$$\log(0) = x \log\left(\frac{1}{3}\right)$$

Since $\log(0)$ is undefined, we know that the x-intercept does not exist.

Analyzing the Graph

Now that we have found the y-intercept and the x-intercept, we can analyze the graph. Since the function is decreasing as $x$ increases, the graph will be a decreasing curve. The y-intercept is at the point (0, 5), and the x-intercept does not exist.

Conclusion

In conclusion, the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ is a decreasing curve with a y-intercept at the point (0, 5) and no x-intercept. This function is an exponential function, and its behavior is determined by the base of the exponent.

Graph Options

There are several graph options that could represent the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$. However, only one of these graphs accurately represents the function.

Graph 1

Graph 1 is a decreasing curve with a y-intercept at the point (0, 5) and no x-intercept. This graph accurately represents the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$.

Graph 2

Graph 2 is an increasing curve with a y-intercept at the point (0, 5) and an x-intercept at the point (1, 0). This graph does not accurately represent the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$.

Graph 3

Graph 3 is a decreasing curve with a y-intercept at the point (0, 5) and an x-intercept at the point (1, 0). This graph does not accurately represent the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$.

Final Answer

The graph that accurately represents the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ is Graph 1.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-20.
• [2] "Graphing Exponential Functions". Purplemath. Retrieved 2023-02-20.

Additional Resources

• [1] Khan Academy. "Exponential Functions". Retrieved 2023-02-20.
• [2] Mathway. "Graphing Exponential Functions". Retrieved 2023-02-20.
  

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Introduction

In our previous article, we explored the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$. We analyzed the behavior of the function and determined that the graph is a decreasing curve with a y-intercept at the point (0, 5) and no x-intercept. In this article, we will answer some frequently asked questions about the graph of the function.

Q&A

Q: What is the y-intercept of the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$?

A: The y-intercept of the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ is at the point (0, 5).

Q: What is the x-intercept of the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$?

A: The x-intercept of the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ does not exist.

Q: Is the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ increasing or decreasing?

A: The graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ is decreasing.

Q: What is the base of the exponent in the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$?

A: The base of the exponent in the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ is $\frac{1}{3}$.

Q: What is the value of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ when $x = 0$?

A: The value of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ when $x = 0$ is 5.

Q: What is the value of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ when $x = 1$?

A: The value of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ when $x = 1$ is $\frac{5}{3}$.

Q: What is the equation of the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$?

A: The equation of the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$ is $y = 5 \cdot \left(\frac{1}{3}\right)^x$.

Conclusion

In this article, we answered some frequently asked questions about the graph of the function $y = 5 \cdot \left(\frac{1}{3}\right)^x$. We hope that this article has been helpful in understanding the behavior of the function and its graph.

Additional Resources

• [1] Khan Academy. "Exponential Functions". Retrieved 2023-02-20.
• [2] Mathway. "Graphing Exponential Functions". Retrieved 2023-02-20.
• [3] Purplemath. "Graphing Exponential Functions". Retrieved 2023-02-20.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-20.
• [2] "Graphing Exponential Functions". Purplemath. Retrieved 2023-02-20.