How To Find Minimum Of Quadratic Function

Finding the minimum value of a quadratic function is a fundamental concept in algebra and calculus, with applications ranging from physics and engineering to economics and computer science. A quadratic function, characterized by its parabolic shape, always has either a minimum or a maximum value, depending on the sign of its leading coefficient. This article provides a comprehensive guide on how to find the minimum value of a quadratic function, covering various methods and providing detailed explanations for each.

Understanding Quadratic Functions

A quadratic function is generally expressed in the form:

f(x) = ax² + bx + c

Where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola.

• If a > 0, the parabola opens upwards, and the function has a minimum value.
• If a < 0, the parabola opens downwards, and the function has a maximum value.

The minimum or maximum value occurs at the vertex of the parabola. The x-coordinate of the vertex, denoted as h, can be found using the formula:

h = -b / 2a

Once you find the x-coordinate of the vertex, you can substitute it back into the original quadratic function to find the y-coordinate of the vertex, denoted as k. This y-coordinate represents the minimum or maximum value of the function.

k = f(h) = a(-b / 2a)² + b(-b / 2a) + c

Let’s explore several methods to find the minimum value of a quadratic function.

Method 1: Using the Vertex Formula

The vertex formula is the most straightforward method to find the minimum or maximum value of a quadratic function. The vertex of the parabola gives the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value.

Steps:

1. Identify the coefficients:

   • Start with the quadratic function in the standard form: f(x) = ax² + bx + c.
   • Identify the values of a, b, and c.

2. Find the x-coordinate of the vertex (h):

   • Use the formula: h = -b / 2a.
   • Substitute the values of a and b into the formula and calculate h.

3. Find the y-coordinate of the vertex (k):

   • Substitute the value of h back into the original quadratic function: k = f(h) = a(h)² + b(h) + c.
   • Calculate k, which is the minimum or maximum value of the function.

Example:

Find the minimum value of the quadratic function: f(x) = 2x² - 8x + 6

1. Identify the coefficients:

   • a = 2, b = -8, c = 6

2. Find the x-coordinate of the vertex (h):

   • h = -(-8) / (2 * 2) = 8 / 4 = 2

3. Find the y-coordinate of the vertex (k):

   • k = f(2) = 2(2)² - 8(2) + 6 = 2(4) - 16 + 6 = 8 - 16 + 6 = -2

Therefore, the minimum value of the quadratic function f(x) = 2x² - 8x + 6 is -2, and it occurs at x = 2.

Method 2: Completing the Square

Completing the square is another powerful technique to rewrite the quadratic function in vertex form, making it easy to identify the vertex and hence the minimum or maximum value.

Steps:

1. Start with the quadratic function:

   • f(x) = ax² + bx + c

2. Factor out a from the x² and x terms:

   • f(x) = a(x² + (b/a)x) + c

3. Complete the square inside the parentheses:

   • To complete the square, take half of the coefficient of x (which is b/a), square it ((b/2a)²), and add and subtract it inside the parentheses.
   • f(x) = a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c

4. Rewrite the expression as a perfect square:

   • f(x) = a((x + b/2a)² - (b/2a)²) + c

5. Distribute a and simplify:

   • f(x) = a(x + b/2a)² - a(b/2a)² + c
   • f(x) = a(x + b/2a)² - b² / 4a + c

6. Combine the constants:

   • f(x) = a(x + b/2a)² + (4ac - b²) / 4a

Now the quadratic function is in vertex form: f(x) = a(x - h)² + k, where the vertex is (h, k), and h = -b/2a and k = (4ac - b²) / 4a.

Example:

Find the minimum value of the quadratic function: f(x) = x² - 6x + 5

1. Start with the quadratic function:

   • f(x) = x² - 6x + 5

2. Factor out a (in this case, a = 1):

   • f(x) = 1(x² - 6x) + 5

3. Complete the square inside the parentheses:

   • Half of -6 is -3, and (-3)² = 9. Add and subtract 9 inside the parentheses.
   • f(x) = (x² - 6x + 9 - 9) + 5

4. Rewrite the expression as a perfect square:

   • f(x) = (x - 3)² - 9 + 5

5. Simplify:

   • f(x) = (x - 3)² - 4

From the vertex form, f(x) = (x - 3)² - 4, we can see that the vertex is (3, -4). Therefore, the minimum value of the quadratic function is -4, and it occurs at x = 3.

Method 3: Using Calculus (Derivatives)

Calculus provides another method for finding the minimum or maximum value of a quadratic function. This method involves finding the derivative of the function and setting it to zero to find the critical points.

Steps:

1. Start with the quadratic function:

   • f(x) = ax² + bx + c

2. Find the first derivative of the function:

   • f'(x) = 2ax + b

3. Set the first derivative equal to zero and solve for x:

   • 2ax + b = 0
   • x = -b / 2a

4. Find the second derivative of the function:

   • f''(x) = 2a

5. Determine if the critical point is a minimum or maximum:

   • If f''(x) > 0 (i.e., a > 0), the critical point is a minimum.
   • If f''(x) < 0 (i.e., a < 0), the critical point is a maximum.

6. Find the y-coordinate of the vertex:

   • Substitute the value of x back into the original function: f(-b / 2a) = a(-b / 2a)² + b(-b / 2a) + c

Example:

Find the minimum value of the quadratic function: f(x) = 3x² - 12x + 7

1. Start with the quadratic function:

   • f(x) = 3x² - 12x + 7

2. Find the first derivative of the function:

   • f'(x) = 6x - 12

3. Set the first derivative equal to zero and solve for x:

   • 6x - 12 = 0
   • 6x = 12
   • x = 2

4. Find the second derivative of the function:

   • f''(x) = 6

5. Determine if the critical point is a minimum or maximum:

   • Since f''(x) = 6 > 0, the critical point is a minimum.

6. Find the y-coordinate of the vertex:

   • f(2) = 3(2)² - 12(2) + 7 = 3(4) - 24 + 7 = 12 - 24 + 7 = -5

Therefore, the minimum value of the quadratic function f(x) = 3x² - 12x + 7 is -5, and it occurs at x = 2.

Method 4: Graphical Method

The graphical method involves plotting the quadratic function and visually identifying the vertex, which represents the minimum or maximum value.

Steps:

1. Create a table of values:

   • Choose several values of x and calculate the corresponding values of f(x).

2. Plot the points on a graph:

   • Plot the points (x, f(x)) on a coordinate plane.

3. Draw the parabola:

   • Connect the points to form a smooth curve, which is the parabola.

4. Identify the vertex:

   • Visually locate the vertex of the parabola, which is the lowest point (for a minimum) or the highest point (for a maximum).

5. Read the coordinates of the vertex:

   • Determine the x and y coordinates of the vertex. The y-coordinate is the minimum or maximum value of the function.

Example:

Find the minimum value of the quadratic function: f(x) = x² - 4x + 3

1. Create a table of values:

2. Plot the points on a graph:

   • Plot the points (0, 3), (1, 0), (2, -1), (3, 0), and (4, 3) on a coordinate plane.

3. Draw the parabola:

   • Connect the points to form a parabola.

4. Identify the vertex:

   • The vertex of the parabola is at the point (2, -1).

5. Read the coordinates of the vertex:

   • The x-coordinate is 2, and the y-coordinate is -1.

Therefore, the minimum value of the quadratic function f(x) = x² - 4x + 3 is -1, and it occurs at x = 2.

Practical Applications

Finding the minimum or maximum value of a quadratic function has numerous practical applications across various fields.

• Physics: In projectile motion, the maximum height reached by an object can be found using the vertex of a quadratic function that models the object's trajectory.
• Engineering: Optimizing the design of structures to minimize material usage or maximize strength often involves finding the minimum or maximum of a quadratic function.
• Economics: Determining the optimal price point to maximize profit can be modeled using quadratic functions, where the maximum value represents the highest profit.
• Computer Science: In optimization algorithms, quadratic functions are often used to model the objective function, and finding the minimum value helps in finding the optimal solution.

Conclusion

Finding the minimum value of a quadratic function is a fundamental skill with broad applications. Whether you choose to use the vertex formula, complete the square, apply calculus, or utilize graphical methods, understanding these techniques provides valuable tools for solving a wide range of problems. Each method offers a unique approach, and familiarity with all of them enhances your problem-solving capabilities. By mastering these techniques, you can effectively analyze and optimize quadratic functions in various real-world scenarios.