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The vertex form of a quadratic equation is a particularly useful way of expressing the equation of a parabola. It makes it easy to identify the vertex, which is a crucial feature of the parabola. The general vertex form of a quadratic equation is given by:\[ y = a(x-h)^2 + k \]where

• \(a\) dictates the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• \((h, k)\) is the vertex of the parabola.

In our exercise, the equation \[y = -\frac{1}{2}(x-2)^2 - 4\] shows that the vertex \((h, k)\) is \((2, -4)\). This is the highest point on the parabola since the parabola opens downward, as indicated by the negative \(a\). Knowing the vertex gives us a point on the graph and helps in graphing the parabola accurately.

When graphing a parabola, identifying its focus and directrix adds depth to our understanding of its geometric properties. The focus is a fixed point inside the parabola, and the directrix is a line outside the parabola such that every point on the parabola is equidistant from the focus and the directrix.For a parabola expressed in vertex form \[ y = a(x-h)^2 + k \], the focus and directrix can be determined using the formula \(\frac{1}{4|a|}\) to find the focal length.

• The focus for our parabola is \((2, -4.5)\), which is \(\frac{1}{2}\) units below the vertex because the parabola opens downwards.
• The directrix, on the other hand, is the line \(y = -3.5\), which lies \(\frac{1}{2}\) units above the vertex.

This construction is central to the definition of a parabola and enhances the graphical representation by identifying significant features of the graph.

Completing the square is a method used to transform a quadratic equation into vertex form. It involves creating a perfect square trinomial from the quadratic expression.To complete the square for a quadratic \(x^2 - 4x\):

• Start by taking half of the coefficient of \(x\), which is \(-4\),resulting in \(-2\).
• Square \(-2\) resulting in \(4\).
• Add and subtract \(4\) inside the expression gives us a perfect square \((x-2)^2\).

By incorporating this into our equation \(-2(x-2)^2 - 4y = 16\), we effectively change it into a suitable form for graphing the parabola by rewriting it with its vertex form. This step is crucial for simplification and helps us locate the vertex easily, leading to a more straightforward graphing process.

Quadratic equations are polynomial equations with the highest degree of 2. They take the general form:\[ ax^2 + bx + c = 0 \]where:

• \(a\), \(b\), and \(c\) are constants,
• \(x\) is the variable.

The standard form can be manipulated to uncover various features of the graph, such as intercepts, vertex, and axis of symmetry.In our exercise, we began with a quadratic equation in the form \(-2x^2 + 8x - 4y - 24 = 0\).By isolating the quadratic term and completing the square, we transformed it into a form suitable for identifying both the vertex and key geometrical properties like the focus and directrix. Understanding how quadratic equations transform allows us to graph parabolas accurately, showing their symmetry and principal features effectively.