What Is The Vertex Of The Graph Of $f(x)=|x+5|-6$?A. ( − 6 , − 5 (-6, -5 ( − 6 , − 5 ] B. ( − 6 , 5 (-6, 5 ( − 6 , 5 ] C. ( − 5 , − 6 (-5, -6 ( − 5 , − 6 ] D. ( 5 , − 6 (5, -6 ( 5 , − 6 ]

Understanding the Graph of a Function

When analyzing the graph of a function, it's essential to understand the different components that make up the graph. The vertex of a graph is the point where the function changes from increasing to decreasing or vice versa. In the case of the function $f(x)=|x+5|-6$, we need to find the vertex of the graph.

The Absolute Value Function

The absolute value function is defined as $|x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases}$

In the function $f(x)=|x+5|-6$, the absolute value is taken of the expression $x+5$. This means that the function will change its behavior at the point where $x+5$ changes sign.

Finding the Vertex

To find the vertex of the graph, we need to find the point where the function changes from increasing to decreasing or vice versa. This occurs when the derivative of the function is equal to zero.

Calculating the Derivative

To calculate the derivative of the function $f(x)=|x+5|-6$, we need to use the chain rule and the fact that the derivative of the absolute value function is given by:

$$\frac{d}{dx}|x| = \begin{cases} 1, & \text{if } x > 0 \\ -1, & \text{if } x < 0 \end{cases}$$

Applying the Chain Rule

Using the chain rule, we can calculate the derivative of the function $f(x)=|x+5|-6$ as follows:

$$f'(x) = \frac{d}{dx}(|x+5|-6)$$

$$f'(x) = \frac{d}{dx}|x+5| - \frac{d}{dx}6$$

$$f'(x) = \frac{d}{dx}|x+5|$$

Evaluating the Derivative

Since the derivative of the absolute value function is given by:

$$\frac{d}{dx}|x| = \begin{cases} 1, & \text{if } x > 0 \\ -1, & \text{if } x < 0 \end{cases}$$

we can evaluate the derivative of the function $f(x)=|x+5|-6$ as follows:

$$f'(x) = \begin{cases} 1, & \text{if } x+5 > 0 \\ -1, & \text{if } x+5 < 0 \end{cases}$$

Finding the Vertex

To find the vertex of the graph, we need to find the point where the function changes from increasing to decreasing or vice versa. This occurs when the derivative of the function is equal to zero.

Solving for x

Setting the derivative equal to zero, we get:

$$f'(x) = 0$$

$$\begin{cases} 1, & \text{if } x+5 > 0 \\ -1, & \text{if } x+5 < 0 \end{cases} = 0$$

Solving for x+5 > 0

If $x+5 > 0$, then $x > -5$. In this case, the derivative is equal to 1, and we have:

$$1 = 0$$

This is a contradiction, so there is no solution for $x > -5$.

Solving for x+5 < 0

If $x+5 < 0$, then $x < -5$. In this case, the derivative is equal to -1, and we have:

$$-1 = 0$$

This is a contradiction, so there is no solution for $x < -5$.

Conclusion

Since there is no solution for $x > -5$ or $x < -5$, we conclude that the vertex of the graph of $f(x)=|x+5|-6$ does not exist.

However, we can still find the vertex of the graph by analyzing the behavior of the function.

Analyzing the Behavior of the Function

The function $f(x)=|x+5|-6$ is a piecewise function, and its behavior changes at the point where $x+5$ changes sign.

Finding the Vertex

To find the vertex of the graph, we need to find the point where the function changes from increasing to decreasing or vice versa.

Solving for x

Setting $x+5 = 0$, we get:

$$x = -5$$

Finding the y-Coordinate

Substituting $x = -5$ into the function $f(x)=|x+5|-6$, we get:

$$f(-5) = |-5+5|-6$$

$$f(-5) = |-6|-6$$

$$f(-5) = 6-6$$

$$f(-5) = 0$$

Conclusion

The vertex of the graph of $f(x)=|x+5|-6$ is the point $(-5, 0)$.

Answer

The correct answer is C. $(-5, 0)$

Q: What is the vertex of the graph of $f(x)=|x+5|-6$?

A: The vertex of the graph of $f(x)=|x+5|-6$ is the point $(-5, 0)$.

Q: Why is the vertex of the graph important?

A: The vertex of the graph is important because it represents the point where the function changes from increasing to decreasing or vice versa. This point is also known as the minimum or maximum point of the function.

Q: How do I find the vertex of the graph?

A: To find the vertex of the graph, you need to find the point where the function changes from increasing to decreasing or vice versa. This can be done by analyzing the behavior of the function or by using calculus to find the derivative of the function and setting it equal to zero.

Q: What is the derivative of the function $f(x)=|x+5|-6$?

A: The derivative of the function $f(x)=|x+5|-6$ is given by:

$$f'(x) = \begin{cases} 1, & \text{if } x+5 > 0 \\ -1, & \text{if } x+5 < 0 \end{cases}$$

Q: How do I evaluate the derivative of the function?

A: To evaluate the derivative of the function, you need to consider the two cases: when $x+5 > 0$ and when $x+5 < 0$.

Q: What is the significance of the point $x = -5$?

A: The point $x = -5$ is significant because it is the point where $x+5$ changes sign. This point is also known as the vertex of the graph.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, you need to substitute $x = -5$ into the function $f(x)=|x+5|-6$.

Q: What is the y-coordinate of the vertex?

A: The y-coordinate of the vertex is 0.

Q: What is the vertex of the graph of $f(x)=|x+5|-6$?

A: The vertex of the graph of $f(x)=|x+5|-6$ is the point $(-5, 0)$.

Q: Why is the vertex of the graph important in real-world applications?

A: The vertex of the graph is important in real-world applications because it represents the point where the function changes from increasing to decreasing or vice versa. This point is also known as the minimum or maximum point of the function.

Q: How do I use the vertex of the graph in real-world applications?

A: To use the vertex of the graph in real-world applications, you need to analyze the behavior of the function and find the point where the function changes from increasing to decreasing or vice versa.

Q: What are some common real-world applications of the vertex of the graph?

A: Some common real-world applications of the vertex of the graph include:

• Finding the minimum or maximum point of a function
• Analyzing the behavior of a function
• Finding the point of inflection of a function
• Finding the rate of change of a function

Q: How do I find the point of inflection of a function?

A: To find the point of inflection of a function, you need to find the point where the function changes from concave up to concave down or vice versa.

Q: What is the point of inflection of the function $f(x)=|x+5|-6$?

A: The point of inflection of the function $f(x)=|x+5|-6$ is the point $(-5, 0)$.

Q: Why is the point of inflection of the function important?

A: The point of inflection of the function is important because it represents the point where the function changes from concave up to concave down or vice versa.

Q: How do I use the point of inflection of the function in real-world applications?

A: To use the point of inflection of the function in real-world applications, you need to analyze the behavior of the function and find the point where the function changes from concave up to concave down or vice versa.

Q: What are some common real-world applications of the point of inflection of the function?

A: Some common real-world applications of the point of inflection of the function include:

• Finding the minimum or maximum point of a function
• Analyzing the behavior of a function
• Finding the point of inflection of a function
• Finding the rate of change of a function