For the function f(x) = 2x^2 - 8x + 10, find the minimum value. (2022)

For the function f(x) = 2x^2 - 8x + 10, find the minimum value. (2022)

Step 1: Identify the function we are working with, which is f(x) = 2x^2 - 8x + 10.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = 2, b = -8, and c = 10.
Step 3: To find the minimum value of a quadratic function, we can use the formula for the x-coordinate of the vertex, which is x = -b / (2a).
Step 4: Plug in the values of a and b into the formula: x = -(-8) / (2 * 2) = 8 / 4 = 2.
Step 5: Now that we have x = 2, we need to find the minimum value of the function by substituting x back into the function: f(2) = 2(2^2) - 8(2) + 10.
Step 6: Calculate f(2): f(2) = 2(4) - 16 + 10 = 8 - 16 + 10 = 2.
Step 7: Therefore, the minimum value of the function f(x) is 2.

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 – Identifying the vertex of a parabola represented by a quadratic function, which gives the minimum or maximum value.
Calculating Function Values – Evaluating the function at specific points to find the minimum value.