The Graph Represents The Function F ( X ) = 10 ( 2 ) X F(x)=10(2)^x F ( X ) = 10 ( 2 ) X .How Would The Graph Change If The B B B Value In The Equation Is Decreased But Remains Greater Than 1? Check All That Apply.- The Graph Will Begin At A Lower Point On The

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding how they behave is crucial for various applications in science, engineering, and finance. The graph of an exponential function is characterized by its rapid growth or decay, depending on the base value (b) and the exponent (x). In this article, we will explore how the graph of the function $f(x)=10(2)^x$ changes when the b value is decreased but remains greater than 1.

Understanding Exponential Functions

An exponential function is a mathematical function of the form $f(x)=ab^x$, where a is the initial value, b is the base value, and x is the exponent. The base value (b) determines the rate of growth or decay of the function. When b is greater than 1, the function grows rapidly, and when b is between 0 and 1, the function decays rapidly.

The Original Graph

The original graph of the function $f(x)=10(2)^x$ is a rapidly growing curve that starts at the point (0, 10) and increases exponentially as x increases. The base value (b) is 2, which is greater than 1, resulting in a rapid growth of the function.

Decreasing the b Value

When the b value is decreased but remains greater than 1, the graph of the function changes in several ways. Here are the possible changes:

The Graph Will Begin at a Lower Point on the Y-Axis

When the b value is decreased, the initial value of the function (a) remains the same, but the rate of growth is reduced. As a result, the graph begins at a lower point on the y-axis. This is because the function grows more slowly, resulting in a lower initial value.

The Graph Will Grow More Slowly

As the b value decreases, the rate of growth of the function also decreases. This means that the graph will grow more slowly, resulting in a more gradual increase in the y-values.

The Graph Will Eventually Cross the Original Graph

When the b value is decreased, the graph of the function will eventually cross the original graph. This is because the new graph grows more slowly, resulting in a lower y-value at some point.

The Graph Will Have a Lower Asymptote

As the b value decreases, the asymptote of the graph also decreases. This is because the function grows more slowly, resulting in a lower y-value as x approaches infinity.

The Graph Will Have a Greater x-Intercept

When the b value is decreased, the x-intercept of the graph increases. This is because the function grows more slowly, resulting in a greater x-value at which the graph intersects the x-axis.

Conclusion

In conclusion, decreasing the b value in the equation $f(x)=10(2)^x$ results in several changes to the graph of the function. The graph begins at a lower point on the y-axis, grows more slowly, eventually crosses the original graph, has a lower asymptote, and has a greater x-intercept. These changes are a result of the reduced rate of growth of the function, which is caused by the decrease in the b value.

Key Takeaways

• Decreasing the b value in an exponential function results in a more gradual increase in the y-values.
• The graph begins at a lower point on the y-axis when the b value is decreased.
• The graph will eventually cross the original graph when the b value is decreased.
• The graph will have a lower asymptote when the b value is decreased.
• The graph will have a greater x-intercept when the b value is decreased.

Frequently Asked Questions

Q: What happens to the graph when the b value is decreased?

A: The graph begins at a lower point on the y-axis, grows more slowly, eventually crosses the original graph, has a lower asymptote, and has a greater x-intercept.

Q: Why does the graph begin at a lower point on the y-axis?

A: The graph begins at a lower point on the y-axis because the function grows more slowly, resulting in a lower initial value.

Q: What is the effect of decreasing the b value on the asymptote of the graph?

A: Decreasing the b value results in a lower asymptote.

Q: What is the effect of decreasing the b value on the x-intercept of the graph?

A: Decreasing the b value results in a greater x-intercept.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Graphing Exponential Functions" by Khan Academy
• [3] "Exponential Growth and Decay" by Wolfram MathWorld
  Q&A: Understanding the Impact of Decreasing the b Value on the Graph of an Exponential Function =============================================================================================

Introduction

In our previous article, we explored the impact of decreasing the b value on the graph of an exponential function. We discussed how the graph begins at a lower point on the y-axis, grows more slowly, eventually crosses the original graph, has a lower asymptote, and has a greater x-intercept. In this article, we will answer some frequently asked questions about the topic.

Q&A

Q: What is the effect of decreasing the b value on the rate of growth of the function?

A: Decreasing the b value results in a slower rate of growth of the function.

Q: Why does the graph begin at a lower point on the y-axis when the b value is decreased?

A: The graph begins at a lower point on the y-axis because the function grows more slowly, resulting in a lower initial value.

Q: What is the relationship between the b value and the asymptote of the graph?

A: The b value determines the asymptote of the graph. Decreasing the b value results in a lower asymptote.

Q: How does decreasing the b value affect the x-intercept of the graph?

A: Decreasing the b value results in a greater x-intercept.

Q: Can the graph of the function still be exponential if the b value is decreased?

A: Yes, the graph of the function can still be exponential even if the b value is decreased. However, the rate of growth will be slower.

Q: What is the effect of decreasing the b value on the y-intercept of the graph?

A: Decreasing the b value results in a lower y-intercept.

Q: Can the graph of the function cross the original graph if the b value is decreased?

A: Yes, the graph of the function can cross the original graph if the b value is decreased.

Q: What is the relationship between the b value and the rate of decay of the function?

A: The b value determines the rate of decay of the function. Decreasing the b value results in a slower rate of decay.

Q: How does decreasing the b value affect the graph of the function in the long run?

A: Decreasing the b value results in a lower asymptote and a greater x-intercept in the long run.

Conclusion

In conclusion, decreasing the b value in an exponential function results in several changes to the graph of the function. The graph begins at a lower point on the y-axis, grows more slowly, eventually crosses the original graph, has a lower asymptote, and has a greater x-intercept. These changes are a result of the reduced rate of growth of the function, which is caused by the decrease in the b value.

Key Takeaways

• Decreasing the b value results in a slower rate of growth of the function.
• The graph begins at a lower point on the y-axis when the b value is decreased.
• The graph will eventually cross the original graph when the b value is decreased.
• The graph will have a lower asymptote when the b value is decreased.
• The graph will have a greater x-intercept when the b value is decreased.

Frequently Asked Questions

Q: What happens to the graph when the b value is decreased?

A: The graph begins at a lower point on the y-axis, grows more slowly, eventually crosses the original graph, has a lower asymptote, and has a greater x-intercept.

Q: Why does the graph begin at a lower point on the y-axis?

A: The graph begins at a lower point on the y-axis because the function grows more slowly, resulting in a lower initial value.

Q: What is the effect of decreasing the b value on the asymptote of the graph?

A: Decreasing the b value results in a lower asymptote.

Q: What is the effect of decreasing the b value on the x-intercept of the graph?

A: Decreasing the b value results in a greater x-intercept.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Graphing Exponential Functions" by Khan Academy
• [3] "Exponential Growth and Decay" by Wolfram MathWorld