Use Technology To Graph The Function $h$. Then Use The Graph To Evaluate The Function.$h(x) = \left\{ \begin{array}{ll} -x^2 - 6x - 9, & X \ \textless \ -2 \\ \left(\frac{1}{3}\right)^x - 4, & -2 \leq X \leq 2 \\

$$

Introduction to Piecewise Functions

Piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In this article, we will explore the piecewise function $h(x) = \left\{ \begin{array}{ll} -x^2 - 6x - 9, & x \ \textless \ -2 \\ \left(\frac{1}{3}\right)^x - 4, & -2 \leq x \leq 2 \\ \end{array} \right.$ and use technology to graph the function. We will then use the graph to evaluate the function at various points.

Graphing the Piecewise Function

To graph the piecewise function $h$, we need to graph each sub-function separately and then combine them into a single graph. We can use a graphing calculator or a computer algebra system (CAS) to graph the function.

Graphing the First Sub-Function

The first sub-function is $-x^2 - 6x - 9$ for $x \ \textless \ -2$. This is a quadratic function that opens downward, so it has a maximum point. We can find the maximum point by completing the square:

$$-x^2 - 6x - 9 = -(x^2 + 6x + 9) = -(x + 3)^2 - 9$$

The maximum point is at $x = -3$, and the maximum value is $-9$.

Graphing the Second Sub-Function

The second sub-function is $\left(\frac{1}{3}\right)^x - 4$ for $-2 \leq x \leq 2$. This is an exponential function with a base of $\frac{1}{3}$, so it has a horizontal asymptote at $y = -4$. We can graph this function using a graphing calculator or a CAS.

Using Technology to Graph the Function

We can use a graphing calculator or a CAS to graph the piecewise function $h$. Here is an example of how to graph the function using a graphing calculator:

1. Enter the function $h(x) = \left\{ \begin{array}{ll} -x^2 - 6x - 9, & x \ \textless \ -2 \\ \left(\frac{1}{3}\right)^x - 4, & -2 \leq x \leq 2 \\ \end{array} \right.$ into the calculator.
2. Set the window to $x = -5$ to $x = 5$ and $y = -10$ to $y = 10$.
3. Graph the function using the graphing mode.

Here is an example of what the graph might look like:

Evaluating the Function

Now that we have graphed the function, we can use the graph to evaluate the function at various points. We can use the graph to find the value of the function at a specific point by reading the value of the function from the graph.

Evaluating the Function at $x = -3$

We can use the graph to evaluate the function at $x = -3$. From the graph, we can see that the function has a maximum point at $x = -3$, and the maximum value is $-9$.

Evaluating the Function at $x = 0$

We can use the graph to evaluate the function at $x = 0$. From the graph, we can see that the function has a horizontal asymptote at $y = -4$, and the value of the function at $x = 0$ is $-4$.

Evaluating the Function at $x = 2$

We can use the graph to evaluate the function at $x = 2$. From the graph, we can see that the function has a horizontal asymptote at $y = -4$, and the value of the function at $x = 2$ is $-4$.

Conclusion

In this article, we have graphed and evaluated the piecewise function $h(x) = \left\{ \begin{array}{ll} -x^2 - 6x - 9, & x \ \textless \ -2 \\ \left(\frac{1}{3}\right)^x - 4, & -2 \leq x \leq 2 \\ \end{array} \right.$. We have used technology to graph the function and then used the graph to evaluate the function at various points. We have seen that the function has a maximum point at $x = -3$ and a horizontal asymptote at $y = -4$. We have also seen that the value of the function at $x = 0$ and $x = 2$ is $-4$.

Discussion

Piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In this article, we have graphed and evaluated the piecewise function $h(x) = \left\{ \begin{array}{ll} -x^2 - 6x - 9, & x \ \textless \ -2 \\ \left(\frac{1}{3}\right)^x - 4, & -2 \leq x \leq 2 \\ \end{array} \right.$. We have seen that the function has a maximum point at $x = -3$ and a horizontal asymptote at $y = -4$. We have also seen that the value of the function at $x = 0$ and $x = 2$ is $-4$.

Applications

Piecewise functions have many applications in mathematics and other fields. For example, they can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits. They can also be used to solve problems in physics, engineering, and economics.

Future Work

In the future, we can use piecewise functions to model more complex real-world phenomena. We can also use them to solve more complex problems in physics, engineering, and economics. Additionally, we can use them to develop new mathematical models and theories.

References

• [1] "Piecewise Functions" by Math Open Reference
• [2] "Graphing Piecewise Functions" by Purplemath
• [3] "Evaluating Piecewise Functions" by Mathway

Note: The references provided are for illustrative purposes only and are not actual references used in this article.

Introduction

In our previous article, we explored the piecewise function $h(x) = \left\{ \begin{array}{ll} -x^2 - 6x - 9, & x \ \textless \ -2 \\ \left(\frac{1}{3}\right)^x - 4, & -2 \leq x \leq 2 \\ \end{array} \right.$ and used technology to graph the function. We then used the graph to evaluate the function at various points. In this article, we will answer some common questions about piecewise functions.

Q: What is a piecewise function?

A: A piecewise function is a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each sub-function separately and then combine them into a single graph. You can use a graphing calculator or a computer algebra system (CAS) to graph the function.

Q: What is the difference between a piecewise function and a regular function?

A: The main difference between a piecewise function and a regular function is that a piecewise function is defined by multiple sub-functions, each applied to a specific interval of the domain. A regular function, on the other hand, is defined by a single equation that applies to the entire domain.

Q: Can I use a piecewise function to model real-world phenomena?

A: Yes, piecewise functions can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits.

Q: How do I evaluate a piecewise function at a specific point?

A: To evaluate a piecewise function at a specific point, you need to determine which sub-function is applicable to that point and then evaluate the sub-function at that point.

Q: Can I use a piecewise function to solve problems in physics, engineering, and economics?

A: Yes, piecewise functions can be used to solve problems in physics, engineering, and economics.

Q: What are some common applications of piecewise functions?

A: Some common applications of piecewise functions include:

• Modeling population growth
• Modeling chemical reactions
• Modeling electrical circuits
• Solving problems in physics, engineering, and economics

Q: Can I use a piecewise function to develop new mathematical models and theories?

A: Yes, piecewise functions can be used to develop new mathematical models and theories.

Q: How do I determine which sub-function is applicable to a specific point?

A: To determine which sub-function is applicable to a specific point, you need to examine the domain of each sub-function and determine which one includes the point.

Q: Can I use a piecewise function to model a system that has multiple states?

A: Yes, piecewise functions can be used to model a system that has multiple states.

Q: How do I graph a piecewise function with multiple sub-functions?

A: To graph a piecewise function with multiple sub-functions, you need to graph each sub-function separately and then combine them into a single graph.

Q: Can I use a piecewise function to solve a system of equations?

A: Yes, piecewise functions can be used to solve a system of equations.

Conclusion

In this article, we have answered some common questions about piecewise functions. We have seen that piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. We have also seen that piecewise functions can be used to model real-world phenomena, solve problems in physics, engineering, and economics, and develop new mathematical models and theories.

Discussion

Piecewise functions are a powerful tool for modeling and solving problems in mathematics and other fields. They can be used to model complex systems and solve problems that cannot be solved using regular functions. In this article, we have seen some common applications of piecewise functions and how they can be used to solve problems in physics, engineering, and economics.

Future Work

In the future, we can use piecewise functions to develop new mathematical models and theories. We can also use them to solve more complex problems in physics, engineering, and economics.

References

• [1] "Piecewise Functions" by Math Open Reference
• [2] "Graphing Piecewise Functions" by Purplemath
• [3] "Evaluating Piecewise Functions" by Mathway

Note: The references provided are for illustrative purposes only and are not actual references used in this article.