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)

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Introduction

In mathematics, the vertex of a graph is a crucial concept that helps us understand the behavior of a function. It is the point on the graph where the function changes from increasing to decreasing or vice versa. In this article, we will focus on finding the vertex of the graph of the function $g(x) = |x-8| + 6$. We will explore the concept of absolute value functions, identify the vertex, and provide a step-by-step solution to the problem.

What is an Absolute Value Function?

An absolute value function is a type of function that involves the absolute value of a variable or expression. It is denoted by the symbol | | and is defined as:

∣x∣={x,if x≥0−x,if x<0|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

The absolute value function has a unique property: it always returns a non-negative value. This means that the graph of an absolute value function will always be above the x-axis.

The Graph of $g(x) = |x-8| + 6$

To find the vertex of the graph of $g(x) = |x-8| + 6$, we need to understand the behavior of the absolute value function. The graph of an absolute value function is a V-shaped graph, with the vertex at the point where the function changes from increasing to decreasing or vice versa.

In this case, the absolute value function is $|x-8|$, which has a vertex at the point (8, 0). The graph of this function is a V-shaped graph, with the vertex at (8, 0).

Adding 6 to the Absolute Value Function

When we add 6 to the absolute value function, we get $g(x) = |x-8| + 6$. This shifts the graph of the absolute value function up by 6 units.

Finding the Vertex of the Graph

To find the vertex of the graph of $g(x) = |x-8| + 6$, we need to find the point where the function changes from increasing to decreasing or vice versa. This point is the vertex of the graph.

Since the graph of the absolute value function is a V-shaped graph, the vertex of the graph of $g(x) = |x-8| + 6$ will be at the point where the function changes from increasing to decreasing or vice versa.

Step-by-Step Solution

To find the vertex of the graph of $g(x) = |x-8| + 6$, we can follow these steps:

  1. Identify the vertex of the absolute value function: The vertex of the absolute value function is at the point (8, 0).
  2. Add 6 to the absolute value function: When we add 6 to the absolute value function, we get $g(x) = |x-8| + 6$.
  3. Find the vertex of the graph: The vertex of the graph of $g(x) = |x-8| + 6$ will be at the point where the function changes from increasing to decreasing or vice versa.

Conclusion

In this article, we have explored the concept of absolute value functions and identified the vertex of the graph of the function $g(x) = |x-8| + 6$. We have also provided a step-by-step solution to the problem.

The vertex of the graph of $g(x) = |x-8| + 6$ is at the point (8, 6). This is because the graph of the absolute value function is a V-shaped graph, with the vertex at (8, 0), and adding 6 to the absolute value function shifts the graph up by 6 units.

Answer

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

Final Thoughts

Introduction

In our previous article, we explored the concept of absolute value functions and identified the vertex of the graph of the function $g(x) = |x-8| + 6$. In this article, we will answer some frequently asked questions about the vertex of a graph.

Q&A

Q: What is the vertex of a graph?

A: The vertex of a graph is the point where the function changes from increasing to decreasing or vice versa. It is the minimum or maximum point of the graph.

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

A: To find the vertex of a graph, you need to identify the point where the function changes from increasing to decreasing or vice versa. This can be done by finding the minimum or maximum point of the graph.

Q: What is the difference between the vertex and the axis of symmetry?

A: The vertex and the axis of symmetry are related but distinct concepts. The vertex is the point where the function changes from increasing to decreasing or vice versa, while the axis of symmetry is the line that passes through the vertex and is perpendicular to the x-axis.

Q: Can the vertex of a graph be a point of inflection?

A: Yes, the vertex of a graph can be a point of inflection. A point of inflection is a point where the function changes from concave up to concave down or vice versa.

Q: How do I determine the type of vertex (minimum or maximum) of a graph?

A: To determine the type of vertex (minimum or maximum) of a graph, you need to examine the behavior of the function on either side of the vertex. If the function is decreasing on one side and increasing on the other, the vertex is a minimum. If the function is increasing on one side and decreasing on the other, the vertex is a maximum.

Q: Can the vertex of a graph be a point of discontinuity?

A: No, the vertex of a graph cannot be a point of discontinuity. A point of discontinuity is a point where the function is not defined or is not continuous.

Q: How do I find the vertex of a graph with a quadratic function?

A: To find the vertex of a graph with a quadratic function, you can use the formula: $x = -\frac{b}{2a}$, where a and b are the coefficients of the quadratic function.

Q: Can the vertex of a graph be a point of tangency?

A: No, the vertex of a graph cannot be a point of tangency. A point of tangency is a point where the function is tangent to a line or curve.

Q: How do I determine the axis of symmetry of a graph?

A: To determine the axis of symmetry of a graph, you need to find the line that passes through the vertex and is perpendicular to the x-axis.

Q: Can the vertex of a graph be a point of intersection?

A: No, the vertex of a graph cannot be a point of intersection. A point of intersection is a point where two or more functions intersect.

Conclusion

In this article, we have answered some frequently asked questions about the vertex of a graph. We hope that this article has provided a comprehensive guide to understanding the vertex of a graph.

Final Thoughts

In conclusion, the vertex of a graph is an essential concept in mathematics. It helps us understand the behavior of a function and identify the point where the function changes from increasing to decreasing or vice versa. We hope that this article has provided a helpful resource for understanding the vertex of a graph.

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