What Are The Coordinates Of The Vertex Of The Function $f(x)=x^2+10x-3$?A. \[$(-5, -28)\$\] B. \[$(-5, 28)\$\] C. \[$(5, -28)\$\] D. \[$(5, 28)\$\]

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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)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc 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 how to find the coordinates of the vertex of a quadratic function.

Understanding the Vertex Form

The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the function. To find the vertex, we need to rewrite the function in vertex form. The vertex form is useful because it allows us to easily identify the vertex of the function.

Rewriting the Function in Vertex Form

To rewrite the function f(x)=x2+10x−3f(x) = x^2 + 10x - 3 in vertex form, we need to complete the square. Completing the square involves adding and subtracting a constant term to create a perfect square trinomial.

f(x) = x^2 + 10x - 3
f(x) = (x^2 + 10x + 25) - 25 - 3
f(x) = (x + 5)^2 - 28

Identifying the Vertex

Now that we have rewritten the function in vertex form, we can easily identify the vertex. The vertex is given by the values of hh and kk in the vertex form. In this case, the vertex is (−5,−28)(-5, -28).

Conclusion

In conclusion, the coordinates of the vertex of the function f(x)=x2+10x−3f(x) = x^2 + 10x - 3 are (−5,−28)(-5, -28). This can be verified by rewriting the function in vertex form and identifying the vertex.

Answer

The correct answer is A. (−5,−28)(-5, -28).

Why is this Important?

Understanding how to find the vertex of a quadratic function is important because it allows us to analyze the behavior of the function. The vertex is the maximum or minimum point on the graph of the function, and it can be used to determine the direction of the function.

Real-World Applications

The concept of finding the vertex of a quadratic function has many real-world applications. For example, in physics, the vertex of a quadratic function can be used to model the motion of an object under the influence of gravity. In economics, the vertex of a quadratic function can be used to model the demand for a product.

Tips and Tricks

  • To find the vertex of a quadratic function, rewrite the function in vertex form by completing the square.
  • The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the function.
  • The vertex of a quadratic function is the maximum or minimum point on the graph of the function.

Common Mistakes

  • Not rewriting the function in vertex form before identifying the vertex.
  • Not completing the square correctly when rewriting the function in vertex form.

Conclusion

In this article, we will answer some frequently asked questions about finding 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 from decreasing to increasing.

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

A: To find the vertex of a quadratic function, you need to rewrite the function in vertex form by completing the square. The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the function.

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=−b2ah = -\frac{b}{2a}

k=f(h)k = f(h)

where aa, bb, and cc are the coefficients of the quadratic function.

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

A: To complete the square, you need to add and subtract a constant term to create a perfect square trinomial. The constant term is given by (b2a)2\left(\frac{b}{2a}\right)^2.

Q: What is the difference between the x-coordinate and the y-coordinate of the vertex?

A: The x-coordinate of the vertex is given by h=−b2ah = -\frac{b}{2a}, and the y-coordinate of the vertex is given by k=f(h)k = f(h). The difference between the x-coordinate and the y-coordinate of the vertex is given by k−hk - h.

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. If the coefficient of the squared term is positive, the vertex is a minimum. If the coefficient of the squared term is negative, the vertex is a maximum.

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 need to look at the coefficient of the squared term. If the coefficient is positive, the vertex is a minimum. If the coefficient 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 where the function changes from concave to convex or from convex to concave.

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

A: To find the point of inflection of a quadratic function, you need to find the second derivative of the function and set it equal to zero. The point of inflection is given by the x-coordinate of the point where the second derivative is equal to zero.

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. It is also used to determine the direction of the function.

Q: Can the vertex of a quadratic function be used to model real-world phenomena?

A: Yes, the vertex of a quadratic function can be used to model real-world phenomena. For example, in physics, the vertex of a quadratic function can be used to model the motion of an object under the influence of gravity. In economics, the vertex of a quadratic function can be used to model the demand for a product.

Q: How do I apply the concept of the vertex of a quadratic function to real-world problems?

A: To apply the concept of the vertex of a quadratic function to real-world problems, you need to identify the maximum or minimum point on the graph of the function and use it to determine the direction of the function. You can also use the vertex to model real-world phenomena such as the motion of an object under the influence of gravity or the demand for a product.