The coordinates of the vertex (y, x) (4, 1) are determined by the equation of the parabola in vertex form, which is y = a(x-h)^2 + k. In this equation, (h, k) represents the coordinates of the vertex. Since the vertex is at (4, 1), the coordinates of the vertex are (4, 1). This means that the parabola is shifted 4 units to the right and 1 unit up from the origin. **

To calculate the vertex form of a quadratic equation, you first need to have the equation in standard form, which is \(y = ax^2 + bx + c\). Then, you can use the formula \(y = a(x-h)^2 + k\) to convert it to vertex form, where \((h, k)\) represents the vertex of the parabola. To find the vertex, you can use the formula \(h = -\frac{b}{2a}\) and \(k = f(h)\), where \(f(h)\) is the value of the function at the x-coordinate of the vertex. **

The vertex form of a quadratic equation is given by y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. The vertex is the point on the parabola where it changes direction, either from opening upwards (if a > 0) or downwards (if a < 0). The values of h and k in the vertex form represent the x-coordinate and y-coordinate of the vertex, respectively. This form allows us to easily identify the vertex and the direction of the parabola without having to graph the equation. **

The vertex of a parabola with the vertex (4, ...) is located at the point (4, ...). The x-coordinate of the vertex remains the same as the given vertex, while the y-coordinate can vary depending on the specific equation of the parabola. The vertex is the point where the parabola changes direction and is the minimum or maximum point of the parabolic curve. **

The general vertex form of a quadratic function is written as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The vertex form of a quadratic function is written as \( y = a(x-h)^2 + k \), where \( a \), \( h \), and \( k \) are constants representing the vertex of the parabola. The main difference between the two forms is that the general vertex form does not explicitly show the vertex of the parabola, while the vertex form directly provides the coordinates of the vertex. **

To convert the standard form of a quadratic function into vertex form when a = 1, you can complete the square. Start by writing the standard form equation as y = x^2 + bx + c. Then, find the value that completes the square by taking half of the coefficient of x and squaring it, which is (b/2)^2. Add and subtract this value inside the parentheses of x^2 + bx to create a perfect square trinomial. Finally, simplify the equation to vertex form by factoring the perfect square trinomial and combining like terms. **

The vertex form of a quadratic equation is written as y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex of the parabola. This form allows us to easily identify the vertex and the direction of the parabola's opening. The parameter 'a' determines the direction and width of the parabola, while (h, k) gives the vertex's position on the coordinate plane. The vertex form is useful for graphing quadratic equations and solving optimization problems. **

A dark vertex is a term used in graph theory to describe a vertex that is not adjacent to any other vertex in the graph. In other words, a dark vertex is isolated and not connected to any other vertex in the graph. This can be visualized as a single point in the graph with no edges connecting it to any other points. Dark vertices are also sometimes referred to as isolated vertices. **