Select The Correct Answer.What Is The Equation Of The Axis Of Symmetry For This Function?$\[ H(x) = -x^2 + 10x - 11 \\]A. \[$ X = -10 \$\]B. \[$ X = -5 \$\]C. \[$ X = 10 \$\]D. \[$ X = 5 \$\]
The axis of symmetry is a fundamental concept in mathematics, particularly in the study of quadratic functions. It is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts. In this article, we will explore the equation of the axis of symmetry for a given quadratic function and provide a step-by-step guide on how to select the correct answer.
What is the Axis of Symmetry?
The axis of symmetry is a line that passes through the vertex of a parabola, which is the point where the parabola changes direction. It is a vertical line that divides the parabola into two symmetrical parts. The equation of the axis of symmetry is given by the formula:
x = -b / 2a
where a and b are the coefficients of the quadratic function.
The Given Quadratic Function
The given quadratic function is:
h(x) = -x^2 + 10x - 11
To find the equation of the axis of symmetry, we need to identify the values of a and b in the function.
Identifying the Coefficients
In the given function, h(x) = -x^2 + 10x - 11, we can see that:
a = -1 (coefficient of x^2) b = 10 (coefficient of x)
Finding the Equation of the Axis of Symmetry
Now that we have identified the values of a and b, we can use the formula to find the equation of the axis of symmetry:
x = -b / 2a x = -10 / (2 * -1) x = -10 / -2 x = 5
Therefore, the equation of the axis of symmetry for the given function is x = 5.
Selecting the Correct Answer
Based on our calculation, the correct answer is:
D. x = 5
The other options are incorrect because they do not match the equation of the axis of symmetry that we calculated.
Conclusion
In conclusion, the axis of symmetry is a fundamental concept in mathematics that plays a crucial role in the study of quadratic functions. By understanding the equation of the axis of symmetry, we can analyze and solve quadratic functions with ease. In this article, we provided a step-by-step guide on how to find the equation of the axis of symmetry for a given quadratic function and selected the correct answer.
Frequently Asked Questions
Q: What is the axis of symmetry? A: The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts.
Q: How do I find the equation of the axis of symmetry? A: To find the equation of the axis of symmetry, you need to identify the values of a and b in the quadratic function and use the formula x = -b / 2a.
Q: What is the significance of the axis of symmetry? A: The axis of symmetry is significant because it helps us analyze and solve quadratic functions. It also helps us understand the behavior of the function and its vertex.
Additional Resources
For more information on the axis of symmetry and quadratic functions, please refer to the following resources:
- Khan Academy: Quadratic Functions
- Mathway: Axis of Symmetry
- Wolfram Alpha: Quadratic Functions
In this article, we will provide answers to frequently asked questions about the axis of symmetry, a fundamental concept in mathematics, particularly in the study of quadratic functions.
Q: What is the axis of symmetry?
A: The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts.
Q: How do I find the equation of the axis of symmetry?
A: To find the equation of the axis of symmetry, you need to identify the values of a and b in the quadratic function and use the formula x = -b / 2a.
Q: What is the significance of the axis of symmetry?
A: The axis of symmetry is significant because it helps us analyze and solve quadratic functions. It also helps us understand the behavior of the function and its vertex.
Q: Can the axis of symmetry be a horizontal line?
A: No, the axis of symmetry is always a vertical line. It is a line that passes through the vertex of a parabola, dividing it into two symmetrical parts.
Q: Can the axis of symmetry be a diagonal line?
A: No, the axis of symmetry is always a vertical line. It is a line that passes through the vertex of a parabola, dividing it into two symmetrical parts.
Q: How do I determine the vertex of a parabola?
A: To determine the vertex of a parabola, you need to find the x-coordinate of the vertex using the formula x = -b / 2a. Then, you can find the y-coordinate of the vertex by plugging the x-coordinate into the quadratic function.
Q: Can the axis of symmetry be a line that passes through the x-intercepts of a parabola?
A: No, the axis of symmetry is a line that passes through the vertex of a parabola, not the x-intercepts.
Q: Can the axis of symmetry be a line that passes through the y-intercept of a parabola?
A: No, the axis of symmetry is a line that passes through the vertex of a parabola, not the y-intercept.
Q: How do I use the axis of symmetry to solve quadratic equations?
A: To use the axis of symmetry to solve quadratic equations, you need to find the equation of the axis of symmetry and then use it to find the x-coordinate of the vertex. Then, you can plug the x-coordinate into the quadratic function to find the y-coordinate of the vertex.
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. By finding the equation of the axis of symmetry, you can determine the relationship between the variables in the system of equations.
Q: Are there any real-world applications of the axis of symmetry?
A: Yes, the axis of symmetry has many real-world applications, including physics, engineering, and economics. It is used to model and analyze real-world phenomena, such as the motion of objects, the behavior of electrical circuits, and the growth of populations.
Conclusion
In conclusion, the axis of symmetry is a fundamental concept in mathematics that plays a crucial role in the study of quadratic functions. By understanding the axis of symmetry, you can analyze and solve quadratic functions with ease. We hope that this Q&A article has provided you with a better understanding of the axis of symmetry and its applications.
Frequently Asked Questions (FAQs)
- What is the axis of symmetry?
- How do I find the equation of the axis of symmetry?
- What is the significance of the axis of symmetry?
- Can the axis of symmetry be a horizontal line?
- Can the axis of symmetry be a diagonal line?
- How do I determine the vertex of a parabola?
- Can the axis of symmetry be a line that passes through the x-intercepts of a parabola?
- Can the axis of symmetry be a line that passes through the y-intercept of a parabola?
- How do I use the axis of symmetry to solve quadratic equations?
- Can the axis of symmetry be used to solve systems of equations?
- Are there any real-world applications of the axis of symmetry?
Additional Resources
For more information on the axis of symmetry and quadratic functions, please refer to the following resources:
- Khan Academy: Quadratic Functions
- Mathway: Axis of Symmetry
- Wolfram Alpha: Quadratic Functions
By understanding the axis of symmetry and quadratic functions, you can develop a deeper appreciation for mathematics and its applications in real-world problems.