The Standard Form Of An Absolute Value Function Is F ( X ) = A ∣ X − H ∣ + K F(x)=a|x-h|+k F ( X ) = A ∣ X − H ∣ + K . Which Of The Following Represents The Vertex?A. ( H , K (h, K ( H , K ] B. ( − H , K (-h, K ( − H , K ] C. ( − K , H (-k, H ( − K , H ] D. ( K , H (k, H ( K , H ]

Introduction

In mathematics, an absolute value function is a type of function that represents the distance of a number from zero on the number line. The standard form of an absolute value function is given by the equation $f(x)=a|x-h|+k$, where $a$, $h$, and $k$ are constants. In this article, we will discuss the standard form of an absolute value function and identify the vertex of the function.

Understanding the Standard Form of an Absolute Value Function

The standard form of an absolute value function is $f(x)=a|x-h|+k$. This equation represents a function that has a vertex at the point $(h, k)$. The absolute value function is a V-shaped graph that opens upwards or downwards, depending on the value of $a$. If $a$ is positive, the graph opens upwards, and if $a$ is negative, the graph opens downwards.

Identifying the Vertex of an Absolute Value Function

The vertex of an absolute value function is the point at which the function changes direction. In the standard form of an absolute value function, the vertex is represented by the point $(h, k)$. This means that the x-coordinate of the vertex is $h$, and the y-coordinate of the vertex is $k$.

Analyzing the Options

Now that we have identified the vertex of an absolute value function, let's analyze the options given in the problem.

• Option A: $(h, k)$. This option represents the vertex of the function, which is the point at which the function changes direction.
• Option B: $(-h, k)$. This option represents a point that is symmetric to the vertex about the y-axis, but it is not the vertex itself.
• Option C: $(-k, h)$. This option represents a point that is symmetric to the vertex about the x-axis, but it is not the vertex itself.
• Option D: $(k, h)$. This option represents a point that is symmetric to the vertex about the y-axis, but it is not the vertex itself.

Conclusion

In conclusion, the standard form of an absolute value function is $f(x)=a|x-h|+k$, and the vertex of the function is represented by the point $(h, k)$. This means that the x-coordinate of the vertex is $h$, and the y-coordinate of the vertex is $k$. Therefore, the correct answer is Option A: $(h, k)$.

Key Takeaways

• The standard form of an absolute value function is $f(x)=a|x-h|+k$.
• The vertex of an absolute value function is represented by the point $(h, k)$.
• The x-coordinate of the vertex is $h$, and the y-coordinate of the vertex is $k$.

Frequently Asked Questions

• What is the standard form of an absolute value function?
  • The standard form of an absolute value function is $f(x)=a|x-h|+k$.
• What is the vertex of an absolute value function?
  • The vertex of an absolute value function is represented by the point $(h, k)$.
• What are the x and y coordinates of the vertex?
  • The x-coordinate of the vertex is $h$, and the y-coordinate of the vertex is $k$.

References

• [1] "Absolute Value Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/absolutevalue.html
• [2] "Vertex Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola.htm

Additional Resources

• Khan Academy: Absolute Value Functions
• Mathway: Absolute Value Functions
• Wolfram Alpha: Absolute Value Functions
  Absolute Value Function Q&A =============================

Q1: What is the standard form of an absolute value function?

A1: The standard form of an absolute value function is $f(x)=a|x-h|+k$, where $a$, $h$, and $k$ are constants.

Q2: What is the vertex of an absolute value function?

A2: The vertex of an absolute value function is represented by the point $(h, k)$.

Q3: What are the x and y coordinates of the vertex?

A3: The x-coordinate of the vertex is $h$, and the y-coordinate of the vertex is $k$.

Q4: What happens to the graph of an absolute value function if $a$ is positive?

A4: If $a$ is positive, the graph of the absolute value function opens upwards.

Q5: What happens to the graph of an absolute value function if $a$ is negative?

A5: If $a$ is negative, the graph of the absolute value function opens downwards.

Q6: How do you find the vertex of an absolute value function?

A6: To find the vertex of an absolute value function, you need to identify the values of $h$ and $k$ in the standard form of the function.

Q7: What is the significance of the vertex in an absolute value function?

A7: The vertex of an absolute value function represents the point at which the function changes direction.

Q8: Can the vertex of an absolute value function be a point of symmetry?

A8: Yes, the vertex of an absolute value function can be a point of symmetry, but it is not necessarily a point of symmetry.

Q9: How do you determine the direction of the graph of an absolute value function?

A9: To determine the direction of the graph of an absolute value function, you need to examine the value of $a$ in the standard form of the function.

Q10: Can an absolute value function have multiple vertices?

A10: No, an absolute value function can only have one vertex.

Q11: What is the relationship between the vertex and the axis of symmetry of an absolute value function?

A11: The vertex of an absolute value function is the point of symmetry about the axis of symmetry.

Q12: Can the axis of symmetry of an absolute value function be a vertical line?

A12: Yes, the axis of symmetry of an absolute value function can be a vertical line.

Q13: How do you find the axis of symmetry of an absolute value function?

A13: To find the axis of symmetry of an absolute value function, you need to identify the value of $h$ in the standard form of the function.

Q14: What is the significance of the axis of symmetry in an absolute value function?

A14: The axis of symmetry of an absolute value function represents the line of symmetry about which the graph is reflected.

Q15: Can the axis of symmetry of an absolute value function be a horizontal line?

A15: No, the axis of symmetry of an absolute value function cannot be a horizontal line.

References

• [1] "Absolute Value Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/absolutevalue.html
• [2] "Vertex Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola.htm

Additional Resources

• Khan Academy: Absolute Value Functions
• Mathway: Absolute Value Functions
• Wolfram Alpha: Absolute Value Functions