Final answer:
The minimum value of the quadratic function h(x) = x² − 16x + 60 is found by completing the square, resulting in h(x) = (x - 8)² - 4. The minimum value occurs at the vertex of the parabola, which is -4 when x = 8.
Step-by-step explanation:
The function given is h(x) = x² − 16x + 60. This is a quadratic equation, and to find its minimum value, we need to complete the square or find the vertex of the parabola that this quadratic equation represents. The vertex form of a quadratic equation is y = a(x - h) ² + k, where (h, k) is the vertex of the parabola. In this case:
h(x) = x² − 16x + 60
First, we factor out any coefficient of x², which is 1 here (so it doesn't change anything), and then find the square of half the coefficient of x, which is (16/2)² = 64. So, we add and subtract 64 inside the equation.
h(x) = (x² − 16x + 64) - 64 + 60
This simplifies to:
h(x) = (x − 8)² - 4
Now, we can see that the minimum value of h(x) occurs when (x − 8)² is zero, which happens when x = 8. Therefore, the minimum value of h(x) is -4.