For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the minimum poin

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Question: For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the minimum point. (2019)

Options:

Correct Answer: (2, -5)

Exam Year: 2019

Solution:

The vertex is at x = -(-12)/(2*3) = 2. The minimum value is f(2) = 3(2^2) - 12(2) + 7 = -5. Thus, the coordinates are (2, -5).

For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the minimum point. (2019)

For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the minimum point. (2019)

Step 1: Identify the function f(x) = 3x^2 - 12x + 7.
Step 2: Recognize that this is a quadratic function in the form f(x) = ax^2 + bx + c, where a = 3, b = -12, and c = 7.
Step 3: To find the x-coordinate of the minimum point (vertex), use the formula x = -b/(2a).
Step 4: Substitute the values of b and a into the formula: x = -(-12)/(2*3).
Step 5: Simplify the expression: x = 12/6 = 2.
Step 6: Now, find the y-coordinate by calculating f(2).
Step 7: Substitute x = 2 into the function: f(2) = 3(2^2) - 12(2) + 7.
Step 8: Calculate 2^2 = 4, then f(2) = 3(4) - 12(2) + 7.
Step 9: Simplify: f(2) = 12 - 24 + 7.
Step 10: Combine the numbers: f(2) = -5.
Step 11: The coordinates of the minimum point are (2, -5).

Quadratic Functions – Understanding the properties of quadratic functions, including how to find the vertex, which represents the minimum or maximum point.
Vertex Formula – Using the vertex formula x = -b/(2a) to find the x-coordinate of the vertex for a quadratic function in standard form.
Function Evaluation – Evaluating the function at the vertex to find the corresponding y-coordinate.