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The vertex of a parabola is a key point that illustrates the pinnacle of its curve, often denoting its highest or lowest point. This can be seen as the turning point of the parabola. To locate it in a quadratic equation of the form \(y = ax^2 + bx + c\), one uses the formula

• \(x = -\frac{b}{2a}\)

As indicated in the exercise above, by substituting \(b = 8\) and \(a = -4\), the x-coordinate of the vertex becomes \(-\frac{8}{2\times(-4)} = 1\).
This x-coordinate helps in finding the y-coordinate by plugging it into the equation itself. So,

• \(y = -4(1)^2 + 8(1) + 2 = 6\)

Thus, the vertex is at the point \((1, 6)\).
This pinpointed intersection showcases the symmetry of the parabola, and it works as a guide as to how the curve opens. If \(a < 0\), as in this case, the parabola opens downwards, and if \(a > 0\), it opens upwards. The vertex thus becomes either the maximum or minimum point.

X-intercepts are the points where the parabola crosses the x-axis. They represent the solutions to the equation when the value of \(y = 0\). For the quadratic equation \(y=-4x^2+8x+2\), x-intercepts are found by solving the equation

• \(-4x^2+8x+2 = 0\)

This can be further broken down using the quadratic formula or factoring, resulting in points

• \(x_1 = 0.5, x_2 = 1\)

These x-intercepts, sometimes referred to as "roots," provide key insights into the graph's behavior, specifically where it intersects with the x-axis. Knowing the intercepts can help sketch the graph of the equation by identifying the horizontal span of the parabola.

The y-intercept is the point at which the graph intersects the y-axis. It provides the starting point of the parabola when \(x = 0\). For our equation \(y=-4x^2+8x+2\), the y-intercept can be pinpointed by simply plugging \(x = 0\) into the equation:

• \(y = -4(0)^2 + 8(0) + 2 = 2\)

Hence, the y-intercept is \(2\), and it's located at the point \((0, 2)\).
This initial point is not only crucial for plotting but also aids in complementing the full picture of how the parabola is shaped along the y-axis.
Having this point identified, we gain a complete sketch by connecting it along with the vertex and x-intercepts.