- The vertex of the quadratic function \(g(x) = (x - 6)^2 + 4\) is \((6, 4)\).
- Since the leading coefficient \(a=1\) is positive, the parabola opens upward, meaning the vertex \((6, 4)\) is a minimum.
- Shifting the graph down by 6 units results in the new function \(g_{new}(x) = (x - 6)^2 - 2\).
- Final answers: Vertex is \(\boxed{(6, 4)}\); it is a \(\boxed{\text{minimum}}\); the new equation is \(\boxed{g(x) = (x - 6)^2 - 2}\).
Explanation
- Identify the vertex of the quadratic function
The function is given by \(g(x) = (x - 6)^2 + 4\). This is in the standard vertex form of a quadratic function, which is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) represents the coordinates of the vertex. By comparing \(g(x) = (x - 6)^2 + 4\) with \(f(x) = a(x - h)^2 + k\), we can identify \(h = 6\) and \(k = 4\). Therefore, the vertex of the graph is \((6, 4)\).
- Determine if the vertex is a maximum or minimum
In the vertex form \(g(x) = a(x - h)^2 + k\), the coefficient \(a\) determines the orientation of the parabola. Here, the expression \((x - 6)^2\) is multiplied by an implicit coefficient of \(a = 1\). Since \(a = 1 > 0\), the parabola opens upward. Because the parabola opens upward, the vertex represents the lowest point on the graph, which is a minimum value of 4 occurring at \(x = 6\).
- Calculate the equation after a vertical shift
A vertical shift of a function \(g(x)\) by \(d\) units downwards is represented by the transformation \(g_{new}(x) = g(x) - d\). In this case, we are shifting the graph down by \(d = 6\) units. Therefore, the new equation is \(g_{new}(x) = g(x) - 6\). Substituting the original expression for \(g(x)\), we get:
\[g_{new}(x) = ((x - 6)^2 + 4) - 6\]
Simplifying this expression:
\[g_{new}(x) = (x - 6)^2 - 2\]
Thus, the equation of the new graph is \(g_{new}(x) = (x - 6)^2 - 2\).
Examples
Understanding quadratic transformations is useful in many real-world fields, such as engineering and physics. For example, if you are designing a parabolic reflector for a satellite dish, changing the vertex position (through vertical or horizontal shifts) directly impacts where the signal is focused. Similarly, in economics, a profit function modeled by a quadratic can be shifted to represent changes in fixed costs or operational adjustments, allowing businesses to analyze how those changes affect their break-even points or maximum profit potential.