What is the minimum value of f(x) = x^2 - 4x + 7? (2023)

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Question: What is the minimum value of f(x) = x^2 - 4x + 7? (2023)

Options:

Correct Answer: 3

Exam Year: 2023

Solution:

The vertex form gives the minimum at x = 2. f(2) = 2, thus minimum value is 3.

What is the minimum value of f(x) = x^2 - 4x + 7? (2023)

What is the minimum value of f(x) = x^2 - 4x + 7? (2023)

Step 1: Identify the function we are working with: f(x) = x^2 - 4x + 7.
Step 2: Recognize that this is a quadratic function in the standard form ax^2 + bx + c, where a = 1, b = -4, and c = 7.
Step 3: Since the coefficient of x^2 (which is a) is positive, the parabola opens upwards, meaning it has a minimum value.
Step 4: To find the x-coordinate of the vertex (which gives the minimum value), use the formula x = -b/(2a).
Step 5: Substitute the values of a and b into the formula: x = -(-4)/(2*1) = 4/2 = 2.
Step 6: Now, substitute x = 2 back into the function to find the minimum value: f(2) = (2)^2 - 4*(2) + 7.
Step 7: Calculate f(2): f(2) = 4 - 8 + 7 = 3.
Step 8: Therefore, the minimum value of f(x) is 3.

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