The Axis Of Symmetry For The Function F ( X ) = − X 2 − 10 X + 16 F(x) = -x^2 - 10x + 16 F ( X ) = − X 2 − 10 X + 16 Is X = − 5 X = -5 X = − 5 . What Are The Coordinates Of The Vertex Of The Graph?A. { (-5, 41)$}$ B. { (-5, 56)$}$ C. { (-5, 76)$}$ D. { (-5, 91)$}$
Introduction
In mathematics, the axis of symmetry is a line that passes through the vertex of a parabola and is perpendicular to the directrix. It is an essential concept in understanding the properties of quadratic functions. In this article, we will explore the relationship between the axis of symmetry and the vertex of a quadratic function, and we will use this knowledge to find the coordinates of the vertex of a given function.
The Axis of Symmetry
The axis of symmetry of a quadratic function is given by the equation x = -b/2a, where a and b are the coefficients of the quadratic function. For the function f(x) = -x^2 - 10x + 16, we have a = -1 and b = -10. Therefore, the axis of symmetry is given by x = -(-10)/2(-1) = -5.
The Vertex of a Quadratic Function
The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is the minimum or maximum point of the graph, depending on the sign of the coefficient of the quadratic term. The vertex of a quadratic function can be found using the formula (h, k) = (-b/2a, f(-b/2a)), where (h, k) are the coordinates of the vertex.
Finding the Coordinates of the Vertex
To find the coordinates of the vertex of the function f(x) = -x^2 - 10x + 16, we need to plug in the value of x = -5 into the function. We have:
f(-5) = -(-5)^2 - 10(-5) + 16 = -25 + 50 + 16 = 41
Therefore, the coordinates of the vertex of the graph are (-5, 41).
Conclusion
In conclusion, the axis of symmetry of a quadratic function is a line that passes through the vertex of the parabola and is perpendicular to the directrix. The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. We have used the formula for the axis of symmetry and the vertex of a quadratic function to find the coordinates of the vertex of the function f(x) = -x^2 - 10x + 16.
Answer
The correct answer is A. (-5, 41).
References
- [1] "Quadratic Functions" by Math Open Reference
- [2] "Axis of Symmetry" by Purplemath
- [3] "Vertex of a Quadratic Function" by Mathway
Additional Information
- The axis of symmetry of a quadratic function is given by the equation x = -b/2a.
- The vertex of a quadratic function can be found using the formula (h, k) = (-b/2a, f(-b/2a)).
- The coordinates of the vertex of a quadratic function can be found by plugging in the value of x = -b/2a into the function.
The Axis of Symmetry and the Vertex of a Quadratic Function: Q&A ================================================================
Q: What is the axis of symmetry of a quadratic function?
A: The axis of symmetry of a quadratic function is a line that passes through the vertex of the parabola and is perpendicular to the directrix. It is given by the equation x = -b/2a, where a and b are the coefficients of the quadratic function.
Q: How do I find the axis of symmetry of a quadratic function?
A: To find the axis of symmetry of a quadratic function, you need to identify the coefficients a and b in the quadratic function. Then, you can plug these values into the equation x = -b/2a to find the axis of symmetry.
Q: What is the vertex of a quadratic function?
A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is the minimum or maximum point of the graph, depending on the sign of the coefficient of the quadratic term.
Q: How do I find the vertex of a quadratic function?
A: To find the vertex of a quadratic function, you need to use the formula (h, k) = (-b/2a, f(-b/2a)), where (h, k) are the coordinates of the vertex. You can plug in the value of x = -b/2a into the function to find the y-coordinate of the vertex.
Q: What is the relationship between the axis of symmetry and the vertex of a quadratic function?
A: The axis of symmetry of a quadratic function passes through the vertex of the parabola. This means that the vertex of the quadratic function lies on the axis of symmetry.
Q: Can the axis of symmetry be vertical or horizontal?
A: The axis of symmetry of a quadratic function can be either vertical or horizontal, depending on the sign of the coefficient of the quadratic term. If the coefficient is positive, the axis of symmetry is vertical. If the coefficient is negative, the axis of symmetry is horizontal.
Q: How do I determine the direction of the axis of symmetry?
A: To determine the direction of the axis of symmetry, you need to look at the sign of the coefficient of the quadratic term. If the coefficient is positive, the axis of symmetry is vertical. If the coefficient is negative, the axis of symmetry is horizontal.
Q: Can the vertex of a quadratic function be a maximum or minimum point?
A: Yes, the vertex of a quadratic function can be either a maximum or minimum point, depending on the sign of the coefficient of the quadratic term. If the coefficient is positive, the vertex is a minimum point. If the coefficient is negative, the vertex is a maximum point.
Q: How do I determine whether the vertex of a quadratic function is a maximum or minimum point?
A: To determine whether the vertex of a quadratic function is a maximum or minimum point, you need to look at the sign of the coefficient of the quadratic term. If the coefficient is positive, the vertex is a minimum point. If the coefficient is negative, the vertex is a maximum point.
Q: Can the axis of symmetry be a single point or a line?
A: The axis of symmetry of a quadratic function is a line that passes through the vertex of the parabola. It is not a single point, but rather a line that extends infinitely in both directions.
Q: How do I find the equation of the axis of symmetry of a quadratic function?
A: To find the equation of the axis of symmetry of a quadratic function, you need to use the equation x = -b/2a, where a and b are the coefficients of the quadratic function.
Q: Can the axis of symmetry be parallel to the x-axis or y-axis?
A: The axis of symmetry of a quadratic function can be parallel to either the x-axis or y-axis, depending on the sign of the coefficient of the quadratic term. If the coefficient is positive, the axis of symmetry is parallel to the y-axis. If the coefficient is negative, the axis of symmetry is parallel to the x-axis.