If the function f(x) = x^2 - 4x + 3 has a minimum value, what is it?

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Question: If the function f(x) = x^2 - 4x + 3 has a minimum value, what is it?

Options:

Correct Answer: -1

Solution:

The vertex form gives the minimum value at x = 2, f(2) = 1.

If the function f(x) = x^2 - 4x + 3 has a minimum value, what is it?

If the function f(x) = x^2 - 4x + 3 has a minimum value, what is it?

Step 1: Identify the function given, which is f(x) = x^2 - 4x + 3.
Step 2: Recognize that this is a quadratic function in the standard form ax^2 + bx + c.
Step 3: Identify the coefficients: a = 1, b = -4, and c = 3.
Step 4: Since a (1) is positive, the parabola opens upwards, meaning it has a minimum value.
Step 5: To find the x-coordinate of the vertex (where the minimum occurs), use the formula x = -b/(2a).
Step 6: Substitute the values: x = -(-4)/(2*1) = 4/2 = 2.
Step 7: Now, find the minimum value by substituting x = 2 back into the function: f(2) = (2)^2 - 4(2) + 3.
Step 8: Calculate f(2): f(2) = 4 - 8 + 3 = -1.
Step 9: Therefore, the minimum value of the function is -1.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find their minimum or maximum values using the vertex form.
Vertex of a Parabola – Identifying the vertex of a parabola represented by a quadratic function, which indicates the minimum or maximum point.
Completing the Square – Using the method of completing the square to convert a quadratic function into vertex form to easily identify the minimum value.