Verified Answer :

  • The quadratic function is given in the standard vertex form: \(f(x) = a(x - h)^2 + k\).
  • For the given function \(f(x) = 3(x - 1)^2 + 4\), the values are \(a = 3\), \(h = 1\), and \(k = 4\).
  • The vertex of the parabola is the point \((h, k)\), which is \((1, 4)\).
  • Final answer: \(\boxed{(1, 4)}\).

Explanation

  1. Identifying the Vertex from the Function Form

To find the vertex of the given quadratic function \(f(x) = 3(x - 1)^2 + 4\), we compare it with the standard vertex form of a quadratic function, which is given by:
\[f(x) = a(x - h)^2 + k\]
In this form, the point \((h, k)\) represents the coordinates of the vertex of the parabola. Identifying the components from our given function:

  • The coefficient \(a\) is \(3\).
  • The value \(h\) is obtained from the expression \((x - 1)\), which implies \(h = 1\).
  • The value \(k\) is added at the end, so \(k = 4\).
  1. Determining the Vertex Coordinates

Comparing the standard form \(f(x) = a(x - h)^2 + k\) with the given function \(f(x) = 3(x - 1)^2 + 4\), we can clearly see the values corresponding to the vertex coordinates.
Matching the parameters, we find:
\[h = 1\]
\[k = 4\]
Thus, the coordinates of the vertex \((h, k)\) are \((1, 4)\).

  1. Conclusion

The vertex of the quadratic function \(f(x) = 3(x - 1)^2 + 4\) is determined by the values \(h\) and \(k\) in its vertex form. By identifying these components, we found the vertex is \((1, 4)\). The final answer is \(\boxed{(1, 4)}\).

Examples

The concept of the vertex of a parabola is fundamental in physics and engineering. For instance, if you throw a ball, its trajectory follows a parabolic path. The vertex of this parabola represents the highest point the ball reaches in the air. By identifying the vertex of the quadratic function describing the height of the ball over time, you can calculate the maximum height reached and the time it takes to get there, which is crucial for predicting performance and safety in sports.

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