Understanding the Concept of a Vertex

Oct 13, 2025 by windailys.com 63 views

Identifying the Function with a Vertex at (2, -9)

Understanding the Concept of a Vertex

In mathematics, a vertex is a point on a curve where the curve changes direction. For a quadratic function, the vertex represents the minimum or maximum point of the curve. The coordinates of the vertex are given by the formula ${(h, k)}$ , where ${h}$ is the x-coordinate and ${k}$ is the y-coordinate.

Analyzing the Options

We are given four quadratic functions and asked to identify which one has a vertex at ${(2, -9)}$ . To determine this, we need to analyze each function and find its vertex.

Option A: ${f(x) = -(x-3)^2}$

The vertex of a quadratic function in the form ${f(x) = a(x-h)^2 + k}$ is given by the point ${(h, k)}$ . In this case, the vertex is ${(3, 0)}$ , which does not match the given vertex ${(2, -9)}$ .

Option B: ${f(x) = (x+8)^2}$

Using the same formula, we can find the vertex of this function. The vertex is given by ${(-8, 0)}$ , which also does not match the given vertex ${(2, -9)}$ .

Option C: ${f(x) = (x-5)(x+1)}$

To find the vertex of this function, we need to expand it and rewrite it in the form ${f(x) = a(x-h)^2 + k}$ . Expanding the function, we get:

${f(x) = x^2 - 4x - 5}$

Rewriting this function in the form ${f(x) = a(x-h)^2 + k}$ , we get:

${f(x) = (x-2)^2 - 9}$

The vertex of this function is ${(2, -9)}$ , which matches the given vertex.

Option D: ${f(x) = -(x-1)(x-5)}$

Expanding this function, we get:

${f(x) = -x^2 + 6x - 5}$

Rewriting this function in the form ${f(x) = a(x-h)^2 + k}$ , we get:

${f(x) = -(x-3)^2 + 4}$

The vertex of this function is ${(3, 4)}$ , which does not match the given vertex ${(2, -9)}$ .

Conclusion

Based on our analysis, we can conclude that the function with a vertex at ${(2, -9)}$ is Option C: ${f(x) = (x-5)(x+1)}$ . This function can be rewritten in the form ${f(x) = a(x-h)^2 + k}$ as ${f(x) = (x-2)^2 - 9}$ , which has a vertex at ${(2, -9)}$ .

Key Takeaways

The vertex of a quadratic function is given by the point ${(h, k)}$ , where ${h}$ is the x-coordinate and ${k}$ is the y-coordinate.
To find the vertex of a quadratic function, we can rewrite it in the form ${f(x) = a(x-h)^2 + k}$ .
The vertex of a quadratic function represents the minimum or maximum point of the curve.

Practice Problems

Find the vertex of the function ${f(x) = (x-2)^2 + 3}$ .
Find the vertex of the function ${f(x) = -(x+1)^2 - 2}$ .
Find the vertex of the function ${f(x) = (x-4)(x+2)}$ .

Solutions

The vertex of the function ${f(x) = (x-2)^2 + 3}$ is ${(2, 3)}$ .
The vertex of the function ${f(x) = -(x+1)^2 - 2}$ is ${(-1, -2)}$ .
The vertex of the function ${f(x) = (x-4)(x+2)}$ is ${(1, -12)}$ .
Vertex of a Quadratic Function: Q&A

Understanding the Concept of a Vertex

Frequently Asked Questions

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is a point on the curve where the curve changes direction. It represents the minimum or maximum point of the curve.

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

A: To find the vertex of a quadratic function, you can rewrite it in the form ${f(x) = a(x-h)^2 + k}$ . The vertex is given by the point ${(h, k)}$ .

Q: What is the formula for finding the vertex of a quadratic function?

A: The formula for finding the vertex of a quadratic function is ${(h, k)}$ , where ${h}$ is the x-coordinate and ${k}$ is the y-coordinate.

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

A: Yes, the vertex of a quadratic function can be a maximum or a minimum point. If the coefficient of the squared term is positive, the vertex is a minimum point. If the coefficient of the squared 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 squared 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 curve where the curve changes from concave up to concave down or vice versa.

Q: How do I find the vertex of a quadratic function in the form ${f(x) = ax^2 + bx + c}$ ?

A: To find the vertex of a quadratic function in the form ${f(x) = ax^2 + bx + c}$ , you can use the formula ${h = -\frac{b}{2a}}$ to find the x-coordinate of the vertex. Then, you can substitute this value into the function to find the y-coordinate of the vertex.

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

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

Practice Problems

Find the vertex of the function ${f(x) = (x-2)^2 + 3}$ .
Find the vertex of the function ${f(x) = -(x+1)^2 - 2}$ .
Find the vertex of the function ${f(x) = (x-4)(x+2)}$ .

Solutions

The vertex of the function ${f(x) = (x-2)^2 + 3}$ is ${(2, 3)}$ .
The vertex of the function ${f(x) = -(x+1)^2 - 2}$ is ${(-1, -2)}$ .
The vertex of the function ${f(x) = (x-4)(x+2)}$ is ${(1, -12)}$ .

Additional Resources

For more information on the vertex of a quadratic function, see the article "Understanding the Concept of a Vertex".
For practice problems and solutions, see the article "Vertex of a Quadratic Function: Practice Problems".
For a list of common quadratic functions and their vertices, see the article "Common Quadratic Functions and Their Vertices".