What Is The Vertex Of The Graph Of $g(x)=|x-8|+6$?A. (6, 8) B. (8, 6) C. (6, -8) D. (-8, 6)

Understanding the Graph of a Function

When dealing with functions, particularly those involving absolute values, it's essential to understand the behavior of the graph. The absolute value function $|x|$ is a V-shaped graph that opens upwards, with its vertex at the origin (0, 0). When we have a function like $g(x) = |x - 8| + 6$, we need to consider how the graph of the absolute value function is shifted and transformed.

Shifting and Transforming the Graph

The graph of $g(x) = |x - 8| + 6$ is a transformation of the graph of $f(x) = |x|$. The absolute value function $|x|$ is shifted 8 units to the right, resulting in $|x - 8|$. This shift is due to the $-8$ inside the absolute value function. Additionally, the entire graph is shifted 6 units upwards, resulting in $|x - 8| + 6$.

Finding the Vertex of the Graph

To find the vertex of the graph of $g(x) = |x - 8| + 6$, we need to consider the vertex of the original graph of $f(x) = |x|$, which is at (0, 0). Since the graph is shifted 8 units to the right, the new x-coordinate of the vertex will be 8. The graph is also shifted 6 units upwards, so the new y-coordinate of the vertex will be 6.

Conclusion

Based on the analysis of the graph of $g(x) = |x - 8| + 6$, we can conclude that the vertex of the graph is at (8, 6). This is because the graph is shifted 8 units to the right and 6 units upwards from the original graph of $f(x) = |x|$.

Answer

The correct answer is B. (8, 6).

Additional Information

It's worth noting that the vertex of a graph can be found using the formula $x = -\frac{b}{2a}$ for quadratic functions in the form $f(x) = ax^2 + bx + c$. However, for functions involving absolute values, we need to consider the transformations of the graph and find the vertex based on the shifted and transformed graph.

Example

Let's consider another example, $g(x) = |x - 2| + 3$. Using the same analysis as before, we can find the vertex of the graph by shifting the original graph of $f(x) = |x|$ 2 units to the right and 3 units upwards. The vertex of the graph will be at (2, 3).

Conclusion

In conclusion, finding the vertex of a graph involving absolute values requires analyzing the transformations of the graph and considering the shifted and transformed graph. By understanding the behavior of the graph and applying the necessary transformations, we can find the vertex of the graph.

Final Answer

The final answer is B. (8, 6).

Frequently Asked Questions

Q1: What is the vertex of the graph of $g(x)=|x-8|+6$?

A1: The vertex of the graph of $g(x)=|x-8|+6$ is at (8, 6).

Q2: How do I find the vertex of a graph involving absolute values?

A2: To find the vertex of a graph involving absolute values, you need to analyze the transformations of the graph and consider the shifted and transformed graph. You can start by finding the vertex of the original graph of $f(x) = |x|$ and then apply the necessary transformations to find the new vertex.

Q3: What is the formula for finding the vertex of a quadratic function?

A3: The formula for finding the vertex of a quadratic function in the form $f(x) = ax^2 + bx + c$ is $x = -\frac{b}{2a}$. However, this formula is not applicable for functions involving absolute values.

Q4: How do I determine the vertex of a graph involving absolute values?

A4: To determine the vertex of a graph involving absolute values, you need to consider the following steps:

• Find the vertex of the original graph of $f(x) = |x|$.
• Apply the necessary shifts and transformations to the graph.
• Find the new vertex based on the shifted and transformed graph.

Q5: What is the vertex of the graph of $g(x)=|x-2|+3$?

A5: The vertex of the graph of $g(x)=|x-2|+3$ is at (2, 3).

Q6: Can I use the formula $x = -\frac{b}{2a}$ to find the vertex of a graph involving absolute values?

A6: No, you cannot use the formula $x = -\frac{b}{2a}$ to find the vertex of a graph involving absolute values. This formula is only applicable for quadratic functions in the form $f(x) = ax^2 + bx + c$.

Q7: How do I find the vertex of a graph involving absolute values with multiple transformations?

A7: To find the vertex of a graph involving absolute values with multiple transformations, you need to apply each transformation step-by-step and find the new vertex after each transformation.

Q8: What is the vertex of the graph of $g(x)=|x-4|+2$?

A8: The vertex of the graph of $g(x)=|x-4|+2$ is at (4, 2).

Q9: Can I use a graphing calculator to find the vertex of a graph involving absolute values?

A9: Yes, you can use a graphing calculator to find the vertex of a graph involving absolute values. However, you need to make sure that the calculator is set to the correct mode and that the graph is displayed in the correct format.

Q10: How do I determine the vertex of a graph involving absolute values with a negative coefficient?

A10: To determine the vertex of a graph involving absolute values with a negative coefficient, you need to consider the following steps:

• Find the vertex of the original graph of $f(x) = |x|$.
• Apply the necessary shifts and transformations to the graph.
• Find the new vertex based on the shifted and transformed graph.
• Consider the effect of the negative coefficient on the graph.

Conclusion

In conclusion, finding the vertex of a graph involving absolute values requires analyzing the transformations of the graph and considering the shifted and transformed graph. By understanding the behavior of the graph and applying the necessary transformations, you can find the vertex of the graph.