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Chapter 3: Problem 14

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)=3 x^{2}-12 x+1$$

Short Answer

Expert verified
The vertex of the parabola represented by the given quadratic function \(f(x) = 3x^{2} - 12x + 1\) is at the point \((2, -3)\).

Step by step solution

01

Finding h (x-coordinate of the vertex)

The x-coordinate of the vertex of a quadratic function is given by \(h = -\frac{b}{2a}\). In this equation our \(a = 3\) and \(b = -12\). So, \(h = -\frac{-12}{2*3} = 2\).
02

Finding k (y-coordinate of the vertex)

The y-coordinate of the vertex of a quadratic function is given by \(k = f(h)\). From our previous step, we found that \(h=2\). Substituting \(x = 2\) into the equation gives us \(k = f(2) = 3*(2)^{2} - 12*2 + 1 = -3\).
03

Writing the vertex

The vertex for a quadratic function is given by the point \((h, k)\). From our calculations, we established that \(h=2\) and \(k=-3\). Therefore, the vertex of the given quadratic function is at the point \((2, -3)\).

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