The Axis Of Symmetry For The Graph Of The Function $f(x)=\frac{1}{4} X^2+b X+10$ Is $x=6$. What Is The Value Of $b$?A. $-12$ B. $-3$ C. $\frac{1}{2}$ D. $3$

Understanding the Axis of Symmetry

The axis of symmetry is a concept in mathematics that plays a crucial role in understanding the behavior of quadratic functions. In the context of quadratic functions, the axis of symmetry is a vertical line that passes through the vertex of the parabola represented by the function. The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

The Given Function and Its Axis of Symmetry

We are given a quadratic function in the form $f(x) = \frac{1}{4}x^2 + bx + 10$, and we are told that the axis of symmetry for the graph of this function is $x = 6$. Using the formula for the axis of symmetry, we can set up an equation to solve for the value of $b$.

Setting Up the Equation

The equation for the axis of symmetry is given by $x = -\frac{b}{2a}$. In this case, we are given that the axis of symmetry is $x = 6$, so we can set up the equation as follows:

$$6 = -\frac{b}{2\left(\frac{1}{4}\right)}$$

Simplifying the Equation

To solve for the value of $b$, we need to simplify the equation. We can start by multiplying both sides of the equation by $2\left(\frac{1}{4}\right)$ to get rid of the fraction:

$$6 \cdot 2\left(\frac{1}{4}\right) = -\frac{b}{2\left(\frac{1}{4}\right)} \cdot 2\left(\frac{1}{4}\right)$$

This simplifies to:

$$3 = -b$$

Solving for $b$

To solve for the value of $b$, we can multiply both sides of the equation by $-1$:

$$-3 = b$$

Therefore, the value of $b$ is $-3$.

Conclusion

In this article, we have discussed the concept of the axis of symmetry in quadratic functions and how it can be used to solve for the value of $b$ in a given quadratic function. We have used the formula for the axis of symmetry to set up an equation and solve for the value of $b$. The value of $b$ is $-3$.

Final Answer

The final answer is $\boxed{-3}$.

Additional Information

• The axis of symmetry is a vertical line that passes through the vertex of the parabola represented by the function.
• The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.
• The value of $b$ can be solved using the formula for the axis of symmetry and the given equation of the axis of symmetry.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved February 2024, from https://www.mathopenref.com/quadratic.html
• [2] "Axis of Symmetry" by Purplemath. Retrieved February 2024, from https://www.purplemath.com/modules/axisym.htm
  The Axis of Symmetry in Quadratic Functions: Q&A =====================================================

Understanding the Axis of Symmetry

The axis of symmetry is a concept in mathematics that plays a crucial role in understanding the behavior of quadratic functions. In the context of quadratic functions, the axis of symmetry is a vertical line that passes through the vertex of the parabola represented by the function. The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q&A Session

Q: What is the axis of symmetry in quadratic functions?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola represented by the function.

Q: How is the equation of the axis of symmetry found?

A: The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the significance of the axis of symmetry in quadratic functions?

A: The axis of symmetry is significant because it helps to determine the vertex of the parabola, which is the highest or lowest point of the parabola.

Q: How is the value of $b$ solved using the axis of symmetry?

A: The value of $b$ can be solved using the formula for the axis of symmetry and the given equation of the axis of symmetry.

Q: What is the formula for the axis of symmetry?

A: The formula for the axis of symmetry is $x = -\frac{b}{2a}$.

Q: What is the significance of the value of $b$ in quadratic functions?

A: The value of $b$ determines the direction and the width of the parabola.

Q: How is the axis of symmetry used in real-world applications?

A: The axis of symmetry is used in various real-world applications, such as physics, engineering, and economics.

Q: Can the axis of symmetry be used to solve quadratic equations?

A: Yes, the axis of symmetry can be used to solve quadratic equations.

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

A: The axis of symmetry passes through the vertex of the parabola.

Q: Can the axis of symmetry be used to determine the maximum or minimum value of a quadratic function?

A: Yes, the axis of symmetry can be used to determine the maximum or minimum value of a quadratic function.

Q: How is the axis of symmetry used in calculus?

A: The axis of symmetry is used in calculus to determine the limits of integration and to solve optimization problems.

Q: Can the axis of symmetry be used to solve systems of equations?

A: Yes, the axis of symmetry can be used to solve systems of equations.

Q: What is the significance of the axis of symmetry in algebra?

A: The axis of symmetry is significant in algebra because it helps to determine the solutions to quadratic equations.

Q: Can the axis of symmetry be used to determine the number of solutions to a quadratic equation?

A: Yes, the axis of symmetry can be used to determine the number of solutions to a quadratic equation.

Q: How is the axis of symmetry used in geometry?

A: The axis of symmetry is used in geometry to determine the properties of shapes and to solve problems involving symmetry.

Q: Can the axis of symmetry be used to solve problems involving optimization?

A: Yes, the axis of symmetry can be used to solve problems involving optimization.

Q: What is the relationship between the axis of symmetry and the graph of a quadratic function?

A: The axis of symmetry is a vertical line that passes through the vertex of the graph of a quadratic function.

Q: Can the axis of symmetry be used to determine the equation of a quadratic function?

A: Yes, the axis of symmetry can be used to determine the equation of a quadratic function.

Q: How is the axis of symmetry used in statistics?

A: The axis of symmetry is used in statistics to determine the properties of distributions and to solve problems involving data analysis.

Q: Can the axis of symmetry be used to solve problems involving probability?

A: Yes, the axis of symmetry can be used to solve problems involving probability.

Q: What is the significance of the axis of symmetry in mathematics education?

A: The axis of symmetry is significant in mathematics education because it helps to develop problem-solving skills and to understand the properties of quadratic functions.

Conclusion

In this article, we have discussed the concept of the axis of symmetry in quadratic functions and its significance in various real-world applications. We have also provided answers to frequently asked questions about the axis of symmetry. The axis of symmetry is a powerful tool that can be used to solve quadratic equations, determine the vertex of a parabola, and solve optimization problems.

Final Answer

The final answer is $\boxed{-3}$.

Additional Information

• The axis of symmetry is a vertical line that passes through the vertex of the parabola represented by the function.
• The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.
• The value of $b$ can be solved using the formula for the axis of symmetry and the given equation of the axis of symmetry.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved February 2024, from https://www.mathopenref.com/quadratic.html
• [2] "Axis of Symmetry" by Purplemath. Retrieved February 2024, from https://www.purplemath.com/modules/axisym.htm