Find the minimum value of f(x) = 4x^2 - 16x + 20. (2022)

Find the minimum value of f(x) = 4x^2 - 16x + 20. (2022)

Step 1: Identify the function you need to analyze, which is f(x) = 4x^2 - 16x + 20.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = 4, b = -16, and c = 20.
Step 3: Determine the x-coordinate of the vertex using the formula x = -b/(2a). Here, b = -16 and a = 4.
Step 4: Calculate -b: -(-16) = 16.
Step 5: Calculate 2a: 2 * 4 = 8.
Step 6: Divide the result from Step 4 by the result from Step 5: x = 16 / 8 = 2.
Step 7: Now, substitute x = 2 back into the function to find the minimum value: f(2) = 4(2^2) - 16(2) + 20.
Step 8: Calculate 2^2: 2^2 = 4.
Step 9: Multiply 4 by 4: 4 * 4 = 16.
Step 10: Multiply -16 by 2: -16 * 2 = -32.
Step 11: Now, combine the results: f(2) = 16 - 32 + 20.
Step 12: Calculate 16 - 32 = -16.
Step 13: Finally, add -16 and 20: -16 + 20 = 4.
Step 14: Therefore, the minimum value of f(x) is 4.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find their minimum or maximum values using the vertex formula.
Vertex of a Parabola – Knowing that the vertex of a parabola given by f(x) = ax^2 + bx + c can be found at x = -b/(2a), which indicates the minimum or maximum point depending on the sign of 'a'.
Function Evaluation – Evaluating the function at the vertex to find the minimum or maximum value.