For the quadratic function y=2x2−12x+21𝑦=2𝑥2−12𝑥+21, fill in the table of values to find the domain and range for the function.

First, let's understand what the domain and range of a function are. The domain of a function is the set of all possible input values (often the "x" variable), while the range is the set of possible output values (often the "y" variable).

For a quadratic function, the domain is all real numbers because you can substitute any real number for x and the function will have a real number for its output.

So, the domain of the function y=2x^2−12x+21 is all real numbers.

The range of a quadratic function depends on the direction of the parabola. If the parabola opens upwards (which is the case when the coefficient of x^2 is positive, like in this function), the range is all numbers greater than or equal to the minimum value of the function. If the parabola opens downwards, the range is all numbers less than or equal to the maximum value of the function.

To find the minimum value of the function, we can complete the square:

y = 2x^2 - 12x + 21 = 2(x^2 - 6x) + 21 = 2(x^2 - 6x + 9) + 21 - 2*9 = 2(x - 3)^2 + 3

So, the minimum value of the function is 3, which occurs when x = 3.

Therefore, the range of the function y=2x^2−12x+21 is y ≥ 3.

To fill in the table of values, you can choose a few values for x and calculate the corresponding values for y using the function. For example:

If x = 0, y = 20^2 - 120 + 21 = 21 If x = 1, y = 21^2 - 121 + 21 = 11 If x = 2, y = 22^2 - 122 + 21 = 5 If x = 3, y = 23^2 - 123 + 21 = 3 If x = 4, y = 24^2 - 124 + 21 = 5 If x = 5, y = 25^2 - 125 + 21 = 11 If x = 6, y = 26^2 - 126 + 21 = 21

And so on. You can choose as many values for x as you want to fill in the table.