To find the domain and range of the parabola with the given equation, we need to understand a few properties of parabolas:
1. A parabola is a set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). Hence, a parabola has a domain of all real numbers, which means it spans from negative infinity to positive infinity.
2. The range of a parabola depends on it opens upwards or downwards. If the coefficient of the x-squared term is positive, the parabola opens upwards, and the y-value of the vertex (the point at which the parabola turns) is the lowest point, it is the minimum y-value. In such a case, the parabola spans from the y-coordinate of the vertex to positive infinity. Conversely, if the coefficient of the x-squared term is negative, the parabola opens downwards, and the y-value of the vertex is the highest point, it is the maximum y-value. In such a case, the parabola spans from negative infinity to the y-coordinate of the vertex.
Given the equation y=0.5(x^(2)-8x-6), let's find the domain and range.
Considering the properties mentioned:
1. Since a parabola corresponds to all real numbers in the domain, our domain will be all real numbers, thus, (-∞, ∞).
2. To find the range, firstly we need to determine if the parabola opens upwards or downwards. This is determined by the coefficient of the x-squared term. As we can see, the coefficient of the x-squared term in the equation given is 0.5, which is a positive number. Therefore, the parabola opens upwards.
3. Now we need to find the y-coordinate of the vertex to determine the range of the parabola. Complete the square to rewrite the equation into vertex form, which is y=a(x-h)²+k, where (h,k) is the vertex.
4. Rewritten in vertex form, we obtain y = 0.5*(x-4)^2-10. The vertex is (4,-10) from the equation.
5. Therefore, since the parabola opens upward and the y-value of the vertex is -10, the range spans from -10 (inclusive) to positive infinity. Thus, the range is [-10, ∞).
In conclusion, the domain of the function is (-∞, ∞), and the range is [-10, ∞).