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

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

Options:

Correct Answer: (2, -1)

Solution:

The vertex is at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 7 = -1. So, the vertex is (2, -1).

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

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

Correct Answer: (2, -1)

Step 1: Identify the coefficients a, b, and c from the function f(x) = 3x^2 - 12x + 7. Here, a = 3, b = -12, and c = 7.
Step 2: Use the formula for the x-coordinate of the vertex, which is x = -b/(2a). Substitute the values of b and a: x = -(-12)/(2*3).
Step 3: Calculate -(-12) which is 12. Then calculate 2*3 which is 6. Now divide 12 by 6: x = 12/6 = 2.
Step 4: Now that we have the x-coordinate of the vertex (x = 2), we need to find the y-coordinate by substituting x back into the function f(x).
Step 5: Substitute x = 2 into f(x): f(2) = 3(2^2) - 12(2) + 7.
Step 6: Calculate 2^2 which is 4. Then calculate 3*4 which is 12. Next, calculate -12*2 which is -24. Now add these values: 12 - 24 + 7.
Step 7: Calculate 12 - 24 which is -12. Then add 7: -12 + 7 = -5.
Step 8: Now we have both coordinates of the vertex: x = 2 and y = -5. Therefore, the vertex is (2, -5).

Quadratic Functions – Understanding the standard form of a quadratic function and how to find its vertex using the formula x = -b/(2a).
Vertex Calculation – Calculating the y-coordinate of the vertex by substituting the x-coordinate back into the function.