For the function f(x) = 3x^2 - 12x + 7, find the minimum value. (2022)

For the function f(x) = 3x^2 - 12x + 7, find the minimum value. (2022)

Step 1: Identify the function you are working with, which is f(x) = 3x^2 - 12x + 7.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = 3, b = -12, and c = 7.
Step 3: To find the minimum value, calculate the x-coordinate of the vertex using the formula x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -(-12)/(2*3) = 12/6 = 2.
Step 5: Now that we have x = 2, substitute this value back into the function to find the minimum value: f(2) = 3(2^2) - 12(2) + 7.
Step 6: Calculate f(2): f(2) = 3(4) - 24 + 7 = 12 - 24 + 7 = -5.
Step 7: The minimum value of the function f(x) is -5.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find the vertex and minimum or maximum values.
Vertex Formula – Using the vertex formula x = -b/(2a) to find the x-coordinate of the vertex for a quadratic function.
Function Evaluation – Evaluating the function at the vertex to find the minimum or maximum value.