Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. \( f(x)=2 x^{2}-28 x+102 \) Complete the square by entering the correct numbers into the expression below. \( f(x)=\square(x-\square)^{2}+\square \)

To complete the square for the expression \( f(x) = 2x^2 - 28x + 102 \), we first factor out the 2 from the quadratic and linear terms: \( f(x) = 2(x^2 - 14x) + 102 \). Next, we need to complete the square inside the parentheses. We take the coefficient of \( x \), which is -14, halve it to get -7, and then square it to get 49. We then rewrite the expression: \( f(x) = 2(x^2 - 14x + 49 - 49) + 102 \). This simplifies to: \( f(x) = 2((x - 7)^2 - 49) + 102 \) \( = 2(x - 7)^2 - 98 + 102 \) \( = 2(x - 7)^2 + 4 \). Thus, we can fill in the squares as follows: \( f(x) = 2(x - 7)^2 + 4 \). --- Now, when it comes to graphing, you'll notice that the vertex form of the quadratic is particularly revealing! The vertex of this parabola is at the point (7, 4), which is the minimum point of the function since the coefficient of the squared term is positive. To graph this function using the technique of shifting, you'll start with the basic parent function \( y = x^2 \) and shift it right by 7 units and up by 4 units. Remember, the parabola will open upwards and be stretched vertically since we have a coefficient of 2 in front. Happy graphing!