Let $f(x) = X^2 + 6x + 11$.What Is The Minimum Value Of The Function?Enter Your Answer In The Box.$\square$

Feb 28, 2025 by ADMIN 108 views

**Finding the Minimum Value of a Quadratic Function**

Introduction

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. In this article, we will focus on finding the minimum value of a quadratic function, specifically the function f(x) = x^2 + 6x + 11.

Understanding Quadratic Functions

Quadratic functions can be represented graphically as a parabola, which is a U-shaped curve. The minimum or maximum value of a quadratic function occurs at the vertex of the parabola. To find the minimum value of a quadratic function, we need to find the x-coordinate of the vertex.

Finding the Vertex of a Quadratic Function

The x-coordinate of 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 the case of the function f(x) = x^2 + 6x + 11, a = 1 and b = 6. Plugging these values into the formula, we get x = -6/2(1) = -6/2 = -3.

Finding the Minimum Value of the Function

Now that we have found the x-coordinate of the vertex, we can find the minimum value of the function by plugging this value into the function. Substituting x = -3 into the function f(x) = x^2 + 6x + 11, we get f(-3) = (-3)^2 + 6(-3) + 11 = 9 - 18 + 11 = 2.

Conclusion

In conclusion, the minimum value of the function f(x) = x^2 + 6x + 11 is 2. This can be found by first finding the x-coordinate of the vertex of the parabola, and then plugging this value into the function.

Step-by-Step Solution

Step 1: Find the x-coordinate of the vertex of the parabola using the formula x = -b/2a.
Step 2: Plug the x-coordinate into the function to find the minimum value.

Example Problems

Find the minimum value of the function f(x) = x^2 + 4x + 4.
Find the minimum value of the function f(x) = x^2 - 2x - 3.

Solutions

For the function f(x) = x^2 + 4x + 4, the x-coordinate of the vertex is x = -4/2(1) = -4/2 = -2. Plugging this value into the function, we get f(-2) = (-2)^2 + 4(-2) + 4 = 4 - 8 + 4 = 0.
For the function f(x) = x^2 - 2x - 3, the x-coordinate of the vertex is x = -(-2)/2(1) = 2/2 = 1. Plugging this value into the function, we get f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4.

Tips and Tricks

To find the minimum value of a quadratic function, you need to find the x-coordinate of the vertex of the parabola.
The x-coordinate of the vertex can be found using the formula x = -b/2a.
Once you have found the x-coordinate of the vertex, plug this value into the function to find the minimum value.

Conclusion

Frequently Asked Questions

Q: What is the minimum value of the function f(x) = x^2 + 5x + 6?

A: To find the minimum value of the function f(x) = x^2 + 5x + 6, we need to find the x-coordinate of the vertex of the parabola. Using the formula x = -b/2a, we get x = -5/2(1) = -5/2 = -2.5. Plugging this value into the function, we get f(-2.5) = (-2.5)^2 + 5(-2.5) + 6 = 6.25 - 12.5 + 6 = -0.25.

Q: How do I find the minimum value of a quadratic function with a negative leading coefficient?

A: To find the minimum value of a quadratic function with a negative leading coefficient, we need to find the x-coordinate of the vertex of the parabola. Using the formula x = -b/2a, we get x = -b/2(-a) = b/2a. Plugging this value into the function, we get f(b/2a) = (b/2a)^2 + b(b/2a) + c = b^2/4a2 + b^2/2a + c = (b^2 + 2ab + 2a^2c)/4a2.

Q: Can I use the quadratic formula to find the minimum value of a quadratic function?

A: Yes, you can use the quadratic formula to find the minimum value of a quadratic function. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. Plugging this value into the function, we get f((-b ± √(b^2 - 4ac)) / 2a) = ((-b ± √(b^2 - 4ac)) / 2a)^2 + b((-b ± √(b^2 - 4ac)) / 2a) + c.

Q: How do I find the minimum value of a quadratic function with a complex leading coefficient?

A: To find the minimum value of a quadratic function with a complex leading coefficient, we need to find the x-coordinate of the vertex of the parabola. Using the formula x = -b/2a, we get x = -b/2a. Plugging this value into the function, we get f(-b/2a) = (-b/2a)^2 + b(-b/2a) + c = b^2/4a2 - b^2/2a + c = (b^2 - 2ab + 2a^2c)/4a2.

Q: Can I use a graphing calculator to find the minimum value of a quadratic function?

A: Yes, you can use a graphing calculator to find the minimum value of a quadratic function. Graph the function and find the x-coordinate of the vertex of the parabola. Then, plug this value into the function to find the minimum value.

Q: How do I find the minimum value of a quadratic function with a rational leading coefficient?

A: To find the minimum value of a quadratic function with a rational leading coefficient, we need to find the x-coordinate of the vertex of the parabola. Using the formula x = -b/2a, we get x = -b/2a. Plugging this value into the function, we get f(-b/2a) = (-b/2a)^2 + b(-b/2a) + c = b^2/4a2 - b^2/2a + c = (b^2 - 2ab + 2a^2c)/4a2.

Q: Can I use a computer algebra system to find the minimum value of a quadratic function?

A: Yes, you can use a computer algebra system to find the minimum value of a quadratic function. Enter the function into the computer algebra system and use the built-in functions to find the minimum value.

Conclusion

In conclusion, finding the minimum value of a quadratic function involves finding the x-coordinate of the vertex of the parabola and then plugging this value into the function. By following the steps outlined in this article, you can find the minimum value of any quadratic function.