Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)

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Question: Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)

Options:

Correct Answer: 4

Exam Year: 2022

Solution:

The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 6.

Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)

Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)

Step 1: Identify the function we need to analyze, 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: Since the coefficient of x^2 (which is a) is positive (2 > 0), the parabola opens upwards, meaning it has a minimum point.
Step 4: To find the x-coordinate of the minimum point, use the formula x = -b / (2a). Here, b = -8 and a = 2.
Step 5: Calculate x = -(-8) / (2 * 2) = 8 / 4 = 2.
Step 6: Now, substitute x = 2 back into the function to find the minimum value: f(2) = 2(2^2) - 8(2) + 10.
Step 7: Calculate f(2) = 2(4) - 16 + 10 = 8 - 16 + 10 = 2.
Step 8: Therefore, the minimum value of the function is 2, which occurs at x = 2.

Quadratic Functions – Understanding the properties of quadratic functions, including finding their minimum or maximum values using the vertex formula.
Vertex of a Parabola – Identifying the vertex of a parabola represented by a quadratic function to determine the minimum or maximum value.
Completing the Square – Using the method of completing the square to rewrite the quadratic function in vertex form.