Graph The Function $y=\left(\frac{1}{2}\right)^{x+2}-9$ Using The Given Table Of Values And Follow The Instructions Below.$\[ \begin{array}{|c|c|} \hline x & Y \\ \hline -10 & 247 \\ \hline -9 & 119 \\ \hline -8 & 55 \\ \hline -7 & 23

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Introduction

Graphing a function using a table of values is a useful technique in mathematics, particularly in algebra and calculus. It allows us to visualize the behavior of a function and make predictions about its behavior at different points. In this article, we will graph the function y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9 using a given table of values.

Understanding the Function

The given function is y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9. This is an exponential function with a base of 12\frac{1}{2} and a horizontal shift of 22 units to the left. The function is also translated vertically by 99 units downward.

Analyzing the Table of Values

The table of values provided is:

x y
-10 247
-9 119
-8 55
-7 23

We can see that as xx increases, yy decreases. This is consistent with the behavior of an exponential function with a base less than 11.

Graphing the Function

To graph the function, we can use the table of values to plot points on a coordinate plane. We will start by plotting the points corresponding to the values in the table.

  • For x=−10x=-10, y=247y=247. Plot the point (−10,247)(-10, 247).
  • For x=−9x=-9, y=119y=119. Plot the point (−9,119)(-9, 119).
  • For x=−8x=-8, y=55y=55. Plot the point (−8,55)(-8, 55).
  • For x=−7x=-7, y=23y=23. Plot the point (−7,23)(-7, 23).

Interpreting the Graph

The graph of the function y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9 is a decreasing exponential curve. As xx increases, the value of yy decreases rapidly. The graph passes through the points (−10,247)(-10, 247), (−9,119)(-9, 119), (−8,55)(-8, 55), and (−7,23)(-7, 23).

Conclusion

Graphing a function using a table of values is a useful technique in mathematics. It allows us to visualize the behavior of a function and make predictions about its behavior at different points. In this article, we graphed the function y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9 using a given table of values.

Key Takeaways

  • Graphing a function using a table of values is a useful technique in mathematics.
  • The function y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9 is a decreasing exponential curve.
  • As xx increases, the value of yy decreases rapidly.

Further Reading

For more information on graphing functions using tables of values, see the following resources:

References

Introduction

In our previous article, we graphed the function y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9 using a table of values. In this article, we will answer some frequently asked questions about graphing this function.

Q&A

Q: What is the base of the exponential function?

A: The base of the exponential function is 12\frac{1}{2}.

Q: What is the horizontal shift of the function?

A: The horizontal shift of the function is 22 units to the left.

Q: What is the vertical translation of the function?

A: The vertical translation of the function is 99 units downward.

Q: How does the function behave as xx increases?

A: As xx increases, the value of yy decreases rapidly.

Q: What is the shape of the graph of the function?

A: The graph of the function is a decreasing exponential curve.

Q: How can I use the table of values to graph the function?

A: You can use the table of values to plot points on a coordinate plane. For each value of xx, calculate the corresponding value of yy and plot the point.

Q: What is the significance of the point (−10,247)(-10, 247) on the graph?

A: The point (−10,247)(-10, 247) is one of the points on the graph that corresponds to a value of xx in the table of values.

Q: How can I determine the behavior of the function at different points?

A: You can use the table of values to determine the behavior of the function at different points. By analyzing the values of yy for different values of xx, you can make predictions about the behavior of the function.

Common Mistakes

  • Mistake 1: Assuming that the function is an increasing exponential curve.
  • Mistake 2: Failing to account for the horizontal shift of the function.
  • Mistake 3: Not using the table of values to plot points on the graph.

Tips and Tricks

  • Tip 1: Use the table of values to plot points on the graph.
  • Tip 2: Analyze the values of yy for different values of xx to make predictions about the behavior of the function.
  • Tip 3: Use the horizontal shift and vertical translation to understand the behavior of the function.

Conclusion

Graphing the function y=(12)x+2−9y=\left(\frac{1}{2}\right)^{x+2}-9 using a table of values is a useful technique in mathematics. By understanding the behavior of the function and using the table of values to plot points on the graph, you can make predictions about the behavior of the function at different points.

Key Takeaways

  • The base of the exponential function is 12\frac{1}{2}.
  • The horizontal shift of the function is 22 units to the left.
  • The vertical translation of the function is 99 units downward.
  • The graph of the function is a decreasing exponential curve.
  • The table of values can be used to plot points on the graph and make predictions about the behavior of the function.

Further Reading

For more information on graphing functions using tables of values, see the following resources:

References