Find the minimum value of f(x) = x^2 - 4x + 7. (2021)

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Question: Find the minimum value of f(x) = x^2 - 4x + 7. (2021)

Options:

Correct Answer: 3

Exam Year: 2021

Solution:

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.

Find the minimum value of f(x) = x^2 - 4x + 7. (2021)

Find the minimum value of f(x) = x^2 - 4x + 7. (2021)

Step 1: Identify the function we need to analyze, which is 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: Understand that the minimum value of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula x = -b/(2a).
Step 4: Calculate the x-coordinate of the vertex: x = -(-4)/(2*1) = 4/2 = 2.
Step 5: Now, substitute x = 2 back into the function to find the minimum value: f(2) = (2)^2 - 4*(2) + 7.
Step 6: Calculate f(2): f(2) = 4 - 8 + 7 = 3.
Step 7: Conclude that the minimum value of f(x) is 3, which occurs at x = 2.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find their minimum or maximum values using vertex form.
Vertex Form – Recognizing that the vertex of a parabola represented by a quadratic function gives the minimum or maximum value.
Completing the Square – Using the method of completing the square to convert a quadratic function into vertex form.