Question

In the quadratic function f(x)=2x²+20x+47, determine the values of h and k.

Answer 1

To find the values of h and k for the quadratic function f(x)=2x²+20x+47, we complete the square to convert it to vertex form; thus, h=-5 and k=-3.

Step-by-step explanation:

The values of h and k in a quadratic function are usually found after converting the function into vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola represented by the quadratic function.

In the quadratic function f(x) = 2x² + 20x + 47, we can complete the square to find the values of h and k.

1. First, factor out the coefficient of the x² term from the x terms: f(x) = 2(x² + 10x) + 47.
2. Next, find the value that completes the square for x² + 10x. This is (10/2)² = 25. Add and subtract this value inside the parenthesis to complete the square, while keeping the equation balanced: f(x) = 2(x² + 10x + 25 - 25) + 47.
3. Now rewrite the equation incorporating the square and simplifying: f(x) = 2((x + 5)² - 25) + 47.
4. Finally, distribute the 2 and simplify to get the vertex form: f(x) = 2(x + 5)² - 50 + 47 which simplifies to f(x) = 2(x + 5)² - 3.

Therefore, the values of h and k are -5 and -3, respectively.