Question

. \(y=-4 x^2+8 x+2\)

Short answer

The vertex of the quadratic equation \(-4x^2+8x+2\) is at \(-1, -4\), the x-intercepts are \(0.5\) and \(1\), and the y-intercept is \(2\).

Step-by-step solution

Identifying the Quadratic Equation

We identify the equation as \(y=-4x^2+8x+2\). It's in the form of a quadratic equation, \(ax^2 + bx + c\), where \(a = -4, b = 8\), and \(c = 2\).

Finding Vertex of the Parabola

We calculate the vertex of the parabola using the formula \(-(b/2a)\). Substituting the values of \(a\) and \(b\), \(-(8/(2*-4)) = -1\). The x-coordinate of the vertex is -1. Substituting this x-coordinate in the equation, the y-coordinate will be \(-4(-1)^2 + 8(-1) + 2 = -4\). Therefore, the vertex of the parabola is \(-1, -4\).

Finding x-Intercepts

The x-intercepts of the parabola are calculated by substituting \(y=0\) in the equation and solving for \(x\). The solutions would be \(x_1 = 0.5, x_2 = 1\).

Finding y-Intercept

The y-intercept is found by substituting \(x = 0\) into the equation: \(y=-4(0)^2 + 8(0) + 2 = 2\). Therefore, the y-intercept is 2.

Key concepts

Vertex of a Parabola

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

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.

Y-Intercept

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.