Select The Correct Texts In The Table.Consider Function { F $} : : : [ f(x) = \begin{cases} -\frac{1}{4} X^2 + 6x + 36, & X \ \textless \ -2 \ 4x - 15, & -2 \leq X \ \textless \ 4 \ 3^{x-4}, & X \ \textgreater \ 4 \end{cases}

Introduction

In this article, we will delve into the world of mathematics and explore a function defined by a piecewise function. The function { f $}$ is given by:

${ f(x) = \begin{cases} -\frac{1}{4} x^2 + 6x + 36, & x \ \textless \ -2 \\ 4x - 15, & -2 \leq x \ \textless \ 4 \\ 3^{x-4}, & x \ \textgreater \ 4 \end{cases} }$

Our goal is to select the correct texts in the table that correspond to the function { f $}$. To achieve this, we need to understand the behavior of the function in different intervals.

Understanding the Function

The function { f $}$ is defined by three different expressions, each corresponding to a specific interval. Let's analyze each expression separately:

Expression 1: $-\frac{1}{4} x^2 + 6x + 36$

This expression is valid for $x < -2$. It represents a quadratic function with a negative leading coefficient, indicating that the parabola opens downwards. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$, where $a = -\frac{1}{4}$ and $b = 6$. Plugging in the values, we get $x = -\frac{6}{-\frac{1}{2}} = 12$. Therefore, the vertex of the parabola is at $x = 12$, which is outside the interval $x < -2$. This means that the parabola is concave downwards and opens towards the left.

Expression 2: $4x - 15$

This expression is valid for $-2 \leq x < 4$. It represents a linear function with a positive slope, indicating that the line is increasing. The y-intercept of the line can be found by plugging in $x = 0$ into the expression, which gives $y = -15$. Therefore, the line passes through the point $(0, -15)$.

Expression 3: $3^{x-4}$

This expression is valid for $x > 4$. It represents an exponential function with a base of $3$, indicating that the function grows rapidly as $x$ increases. The y-intercept of the function can be found by plugging in $x = 0$ into the expression, which gives $y = 3^{-4} = \frac{1}{81}$.

Selecting the Correct Texts

Now that we have analyzed each expression separately, we can select the correct texts in the table that correspond to the function { f $}$. Based on the intervals and expressions, we can conclude that:

• For $x < -2$, the correct text is $-\frac{1}{4} x^2 + 6x + 36$.
• For $-2 \leq x < 4$, the correct text is $4x - 15$.
• For $x > 4$, the correct text is $3^{x-4}$.

Conclusion

In conclusion, we have analyzed the function { f $}$ and selected the correct texts in the table that correspond to the function. We have understood the behavior of the function in different intervals and identified the correct expressions for each interval. This analysis demonstrates the importance of understanding the behavior of functions in different intervals and selecting the correct expressions accordingly.

Mathematical Analysis

Derivative of the Function

To further analyze the function, let's find the derivative of the function using the piecewise definition.

• For $x < -2$, the derivative of $-\frac{1}{4} x^2 + 6x + 36$ is $-\frac{1}{2} x + 6$.
• For $-2 \leq x < 4$, the derivative of $4x - 15$ is $4$.
• For $x > 4$, the derivative of $3^{x-4}$ is $3^{x-4} \ln 3$.

Second Derivative of the Function

To further analyze the function, let's find the second derivative of the function using the piecewise definition.

• For $x < -2$, the second derivative of $-\frac{1}{2} x + 6$ is $-\frac{1}{2}$.
• For $-2 \leq x < 4$, the second derivative of $4$ is $0$.
• For $x > 4$, the second derivative of $3^{x-4} \ln 3$ is $3^{x-4} (\ln 3)^2$.

Graphical Analysis

To further analyze the function, let's graph the function using the piecewise definition.

• For $x < -2$, the graph of $-\frac{1}{4} x^2 + 6x + 36$ is a parabola that opens downwards.
• For $-2 \leq x < 4$, the graph of $4x - 15$ is a line that increases.
• For $x > 4$, the graph of $3^{x-4}$ is an exponential function that grows rapidly.

Conclusion

In conclusion, we have analyzed the function { f $}$ and selected the correct texts in the table that correspond to the function. We have understood the behavior of the function in different intervals and identified the correct expressions for each interval. This analysis demonstrates the importance of understanding the behavior of functions in different intervals and selecting the correct expressions accordingly.

Final Answer

The final answer is:

• For $x < -2$, the correct text is $-\frac{1}{4} x^2 + 6x + 36$.
• For $-2 \leq x < 4$, the correct text is $4x - 15$.
• For $x > 4$, the correct text is $3^{x-4}$.
  Q&A: Selecting the Correct Texts in the Table =============================================

Introduction

In our previous article, we analyzed the function { f $}$ and selected the correct texts in the table that correspond to the function. In this article, we will answer some frequently asked questions related to the function and its analysis.

Q: What is the purpose of the piecewise function?

A: The piecewise function is used to define a function that has different expressions in different intervals. In this case, the function { f $}$ has three different expressions in three different intervals: $x < -2$, $-2 \leq x < 4$, and $x > 4$.

Q: How do I determine which expression to use in each interval?

A: To determine which expression to use in each interval, you need to check the value of $x$ and see which interval it falls into. For example, if $x = -3$, then $x < -2$ and you should use the expression $-\frac{1}{4} x^2 + 6x + 36$. If $x = 2$, then $-2 \leq x < 4$ and you should use the expression $4x - 15$.

Q: What is the derivative of the function?

A: The derivative of the function is a measure of how fast the function changes as $x$ changes. To find the derivative of the function, you need to find the derivative of each expression in each interval and then combine them. The derivative of the function is:

• For $x < -2$, the derivative is $-\frac{1}{2} x + 6$.
• For $-2 \leq x < 4$, the derivative is $4$.
• For $x > 4$, the derivative is $3^{x-4} \ln 3$.

Q: What is the second derivative of the function?

A: The second derivative of the function is a measure of how fast the derivative changes as $x$ changes. To find the second derivative of the function, you need to find the second derivative of each expression in each interval and then combine them. The second derivative of the function is:

• For $x < -2$, the second derivative is $-\frac{1}{2}$.
• For $-2 \leq x < 4$, the second derivative is $0$.
• For $x > 4$, the second derivative is $3^{x-4} (\ln 3)^2$.

Q: How do I graph the function?

A: To graph the function, you need to graph each expression in each interval and then combine them. The graph of the function is:

• For $x < -2$, the graph is a parabola that opens downwards.
• For $-2 \leq x < 4$, the graph is a line that increases.
• For $x > 4$, the graph is an exponential function that grows rapidly.

Q: What is the significance of the function?

A: The function { f $}$ is a piecewise function that has different expressions in different intervals. The function is significant because it demonstrates the importance of understanding the behavior of functions in different intervals and selecting the correct expressions accordingly.

Conclusion

In conclusion, we have answered some frequently asked questions related to the function and its analysis. We have discussed the purpose of the piecewise function, how to determine which expression to use in each interval, and how to graph the function. We have also discussed the significance of the function and its importance in understanding the behavior of functions in different intervals.

Final Answer

The final answer is:

• For $x < -2$, the correct text is $-\frac{1}{4} x^2 + 6x + 36$.
• For $-2 \leq x < 4$, the correct text is $4x - 15$.
• For $x > 4$, the correct text is $3^{x-4}$.

Additional Resources

For more information on the function and its analysis, please refer to the following resources:

• Mathematical Analysis of the Function
• Graphical Analysis of the Function
• Derivative and Second Derivative of the Function

References

• Introduction to Piecewise Functions
• Graphing Piecewise Functions
• Derivatives and Second Derivatives of Piecewise Functions