What Is The $y$-value Of The Vertex Of The Function F ( X ) = − ( X + 8 ) ( X − 14 F(x)=-(x+8)(x-14 F ( X ) = − ( X + 8 ) ( X − 14 ]?A. 121 B. 3 C. 112 D. 6

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Understanding the Problem

To find the $y$-value of the vertex of the function $f(x)=-(x+8)(x-14)$, we need to first identify the type of function and then use the appropriate method to find the vertex. The given function is a quadratic function in the form of $f(x) = a(x - h)(x - k)$, where $(h, k)$ is the vertex of the parabola.

Identifying the Type of Function

The given function $f(x)=-(x+8)(x-14)$ is a quadratic function in the form of $f(x) = a(x - h)(x - k)$. The coefficient of the quadratic term is $-1$, which indicates that the parabola opens downward. This means that the vertex of the parabola will be the maximum point of the function.

Finding the Vertex

To find the vertex of the parabola, we need to find the values of $h$ and $k$ in the equation $f(x) = a(x - h)(x - k)$. In this case, we can rewrite the equation as $f(x) = -(x + 8)(x - 14)$.

Expanding the Equation

To find the values of $h$ and $k$, we need to expand the equation $f(x) = -(x + 8)(x - 14)$. Using the distributive property, we get:

$$f(x) = -(x^2 - 14x + 8x - 112)$$

Simplifying the equation, we get:

$$f(x) = -(x^2 - 6x - 112)$$

Completing the Square

To find the values of $h$ and $k$, we can complete the square by rewriting the equation in the form $f(x) = a(x - h)^2 + k$. To do this, we need to move the constant term to the right-hand side of the equation:

$$f(x) = -x^2 + 6x + 112$$

Finding the Vertex

To find the vertex of the parabola, we need to find the values of $h$ and $k$ in the equation $f(x) = a(x - h)^2 + k$. In this case, we can rewrite the equation as $f(x) = -(x - 3)^2 + 121$.

Identifying the Vertex

The vertex of the parabola is the point $(h, k)$, where $h$ is the value of $x$ that makes the expression $(x - h)^2$ equal to zero, and $k$ is the value of $y$ that makes the expression $a(x - h)^2 + k$ equal to zero. In this case, the vertex is the point $(3, 121)$.

Conclusion

The $y$-value of the vertex of the function $f(x)=-(x+8)(x-14)$ is 121.

Answer

The correct answer is A. 121.

Q: What is the vertex of a quadratic function?

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

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

A: To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can plug the x-coordinate back into the function.

Q: What is the significance of the vertex of a quadratic function?

A: The vertex of a quadratic function is significant because it represents the maximum or minimum point of the function. This point is also known as the turning point of the function.

Q: Can the vertex of a quadratic function be a maximum or a minimum?

A: Yes, the vertex of a quadratic function can be either a maximum or a minimum point. If the coefficient of the quadratic term is positive, the vertex is a minimum point. If the coefficient of the quadratic term is negative, the vertex is a maximum point.

Q: How do I determine whether the vertex of a quadratic function is a maximum or a minimum?

A: To determine whether the vertex of a quadratic function is a maximum or a minimum, you can look at 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 vertex of a quadratic function be a point of inflection?

A: No, the vertex of a quadratic function cannot be a point of inflection. A point of inflection is a point on the graph of a function where the function changes from concave to convex or vice versa. The vertex of a quadratic function is a point of maximum or minimum, not a point of inflection.

Q: How do I find the vertex of a quadratic function in the form $f(x) = a(x - h)^2 + k$?

A: To find the vertex of a quadratic function in the form $f(x) = a(x - h)^2 + k$, you can simply read off the values of $h$ and $k$. The vertex is the point $(h, k)$.

Q: Can the vertex of a quadratic function be a point on the x-axis?

A: Yes, the vertex of a quadratic function can be a point on the x-axis. This occurs when the y-coordinate of the vertex is zero.

Q: How do I find the vertex of a quadratic function that is not in the standard form?

A: To find the vertex of a quadratic function that is not in the standard form, you can first rewrite the function in the standard form by completing the square. Then, you can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.

Q: Can the vertex of a quadratic function be a point on the y-axis?

A: No, the vertex of a quadratic function cannot be a point on the y-axis. The vertex is always a point in the coordinate plane, not a point on the y-axis.

Q: How do I determine whether the vertex of a quadratic function is a local maximum or a local minimum?

A: To determine whether the vertex of a quadratic function is a local maximum or a local minimum, you can look at the coefficient of the quadratic term. If the coefficient is positive, the vertex is a local minimum. If the coefficient is negative, the vertex is a local maximum.

Q: Can the vertex of a quadratic function be a global maximum or a global minimum?

A: Yes, the vertex of a quadratic function can be a global maximum or a global minimum. This occurs when the function is a quadratic function that is bounded above or below by a horizontal line.

Q: How do I find the vertex of a quadratic function that is a global maximum or a global minimum?

A: To find the vertex of a quadratic function that is a global maximum or a global minimum, you can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Then, you can plug the x-coordinate back into the function to find the y-coordinate.

Q: Can the vertex of a quadratic function be a point of discontinuity?

A: No, the vertex of a quadratic function cannot be a point of discontinuity. The vertex is always a point in the coordinate plane, not a point of discontinuity.

Q: How do I determine whether the vertex of a quadratic function is a point of continuity?

A: To determine whether the vertex of a quadratic function is a point of continuity, you can look at the function and determine whether it is continuous at the vertex. If the function is continuous at the vertex, then the vertex is a point of continuity.

Q: Can the vertex of a quadratic function be a point of non-differentiability?

A: No, the vertex of a quadratic function cannot be a point of non-differentiability. The vertex is always a point in the coordinate plane, not a point of non-differentiability.

Q: How do I determine whether the vertex of a quadratic function is a point of differentiability?

A: To determine whether the vertex of a quadratic function is a point of differentiability, you can look at the function and determine whether it is differentiable at the vertex. If the function is differentiable at the vertex, then the vertex is a point of differentiability.

Q: Can the vertex of a quadratic function be a point of inflection?

A: No, the vertex of a quadratic function cannot be a point of inflection. A point of inflection is a point on the graph of a function where the function changes from concave to convex or vice versa. The vertex of a quadratic function is a point of maximum or minimum, not a point of inflection.

Q: How do I determine whether the vertex of a quadratic function is a point of inflection?

A: To determine whether the vertex of a quadratic function is a point of inflection, you can look at the function and determine whether it changes from concave to convex or vice versa at the vertex. If the function changes from concave to convex or vice versa at the vertex, then the vertex is a point of inflection.

Q: Can the vertex of a quadratic function be a point of tangency?

A: No, the vertex of a quadratic function cannot be a point of tangency. A point of tangency is a point on the graph of a function where the function touches a line or curve. The vertex of a quadratic function is a point of maximum or minimum, not a point of tangency.

Q: How do I determine whether the vertex of a quadratic function is a point of tangency?

A: To determine whether the vertex of a quadratic function is a point of tangency, you can look at the function and determine whether it touches a line or curve at the vertex. If the function touches a line or curve at the vertex, then the vertex is a point of tangency.

Q: Can the vertex of a quadratic function be a point of intersection?

A: Yes, the vertex of a quadratic function can be a point of intersection. This occurs when the function intersects with another function at the vertex.

Q: How do I determine whether the vertex of a quadratic function is a point of intersection?

A: To determine whether the vertex of a quadratic function is a point of intersection, you can look at the function and determine whether it intersects with another function at the vertex. If the function intersects with another function at the vertex, then the vertex is a point of intersection.

Q: Can the vertex of a quadratic function be a point of asymptote?

A: No, the vertex of a quadratic function cannot be a point of asymptote. A point of asymptote is a point on the graph of a function where the function approaches a line or curve. The vertex of a quadratic function is a point of maximum or minimum, not a point of asymptote.

Q: How do I determine whether the vertex of a quadratic function is a point of asymptote?

A: To determine whether the vertex of a quadratic function is a point of asymptote, you can look at the function and determine whether it approaches a line or curve at the vertex. If the function approaches a line or curve at the vertex, then the vertex is a point of asymptote.

Q: Can the vertex of a quadratic function be a point of discontinuity?

A: No, the vertex of a quadratic function cannot be a point of discontinuity. The vertex is always a point in the coordinate plane, not a point of discontinuity.

Q: How do I determine whether the vertex of a quadratic function is a point of continuity?

A: To determine whether the vertex of a quadratic function is a point of continuity, you can look at the function and determine whether it is continuous at the vertex. If the function is continuous at the vertex, then the vertex is a point of continuity.

Q: Can the vertex of a quadratic function be a point of non-differentiability?

A: No, the vertex of a quadratic function cannot be a point of non-differentiability. The vertex is always a point in the coordinate plane, not a point of non-differentiability.

Q: How do I determine whether the vertex of a quadratic function is a point of differentiability?

A: To determine whether the vertex of a quadratic function is a point of differentiability, you can look at the function and determine whether it is differentiable at the vertex. If the function is differentiable at