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

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

Find the minimum value of f(x) = x^2 - 4x + 7. (2021) 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: To find the minimum value, we can use the vertex formula. The x-coordinate of the vertex is given by the formula x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -(-4)/(2*1) = 4/2 = 2.
Step 5: Now that we have x = 2, we need to find the corresponding y-value (f(2)).
Step 6: Calculate f(2) by substituting x = 2 into the function: f(2) = 2^2 - 4*2 + 7.
Step 7: Simplify the expression: f(2) = 4 - 8 + 7 = 3.
Step 8: Therefore, the minimum value of the function f(x) is 3.

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.