Which Of The Following Is True About The Vertex Of The Function $f(x)=3(x-2)^2-6$?A. The Vertex Is $(2, -6$\] And It Is A Maximum.B. The Vertex Is $(2, -6$\] And It Is A Minimum.C. The Vertex Is $(-2, -6$\] And It Is A

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. The vertex of a quadratic function is the maximum or minimum point on the graph of the function. In this article, we will discuss the vertex of the function f(x) = 3(x - 2)^2 - 6 and determine which of the given options is true.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function. To find the vertex of the function f(x) = 3(x - 2)^2 - 6, we need to rewrite it in the vertex form.

Rewriting the Function in Vertex Form

To rewrite the function f(x) = 3(x - 2)^2 - 6 in vertex form, we need to expand the squared term and simplify the expression.

f(x) = 3(x - 2)^2 - 6 f(x) = 3(x^2 - 4x + 4) - 6 f(x) = 3x^2 - 12x + 12 - 6 f(x) = 3x^2 - 12x + 6

Now, we can see that the function is in the form f(x) = a(x - h)^2 + k, where a = 3, h = 2, and k = -6.

Determining the Vertex

Since the function is in the vertex form, we can determine the vertex by looking at the values of h and k. The vertex of the function is (h, k) = (2, -6).

Is the Vertex a Maximum or Minimum?

To determine whether the vertex is a maximum or minimum, we need to look at the value of a. If a is positive, the vertex is a minimum. If a is negative, the vertex is a maximum.

In this case, a = 3, which is positive. Therefore, the vertex is a minimum.

Conclusion

Based on the analysis, we can conclude that the vertex of the function f(x) = 3(x - 2)^2 - 6 is (2, -6) and it is a minimum.

Answer

The correct answer is B. The vertex is (2, -6) and it is a minimum.

Additional Information

It's worth noting that the vertex of a quadratic function can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. In this case, a = 3 and b = -12, so x = -(-12) / (2 * 3) = 2.

In the previous article, we discussed the vertex of a quadratic function and how to determine whether it is a maximum or minimum. In this article, we will answer some frequently asked questions about the vertex of a quadratic function.

Q: What is the vertex of a quadratic function?

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

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

A: There are several ways to find the vertex of a quadratic function. One way is to rewrite the function in vertex form, which is f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Another way is to use the formula x = -b / 2a, where a and b are the coefficients of the quadratic 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 on the graph of the function. This point is also known as the turning point, because it is the point where the function changes from increasing to decreasing or vice versa.

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

A: To determine whether the vertex of a quadratic function is a maximum or minimum, you need to look at the value of a. If a is positive, the vertex is a minimum. If a is negative, the vertex is a maximum.

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 x-coordinate of the vertex of a quadratic function?

A: To find the x-coordinate of the vertex of a quadratic function, you can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic function.

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. A point of discontinuity is a point on the graph of a function where the function is not defined. The vertex of a quadratic function is a point on the graph of the function, so it cannot be a point of discontinuity.

Q: How do I graph a quadratic function with a given vertex?

A: To graph a quadratic function with a given vertex, you can use the vertex form of the function, which is f(x) = a(x - h)^2 + k, where (h, k) is the vertex. You can then use the graphing calculator or software to graph the function.

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 is tangent to a line. The vertex of a quadratic function is a point of maximum or minimum, not a point of tangency.

Conclusion

In this article, we answered some frequently asked questions about the vertex of a quadratic function. We hope that this article has provided you with a better understanding of the vertex of a quadratic function and how to determine whether it is a maximum or minimum.