Final answer:
To find a quadratic function based on given values, we solve a system of equations after plugging those values into the standard quadratic form. For the given quadratic equation y = x² + 12x + 27, we use the quadratic formula to find its roots and determine the intervals of positivity. The quadratic formula is a critical component in solving quadratic equations.
Step-by-step explanation:
Finding a Quadratic Function
To find a quadratic function that matches the given set of values (0,8), (2,18), and (4,12), we need to use a standard form of a quadratic equation, which is y = ax² + bx + c. We plug in the points to obtain a system of equations and solve for a, b, and c. After solving, we will have the quadratic function that passes through the given points.
Determining Intervals Where a Quadratic is Positive
To determine the intervals on which the function y = x² + 12x + 27 is positive, we find its roots using the quadratic formula. The formula is x = (-b ± √(b²-4ac))/(2a), applying it to our quadratic equation after identifying values of a, b, and c. Once we have the roots, the sign of the quadratic in each interval determined by these roots will tell us where the function is positive.
The Quadratic Formula
The quadratic formula is used to solve for the roots of any quadratic equation in the form ax² + bx + c = 0. In this context, a, b, and c are known as the coefficients of the quadratic equation, and the solutions reveal the x-intercepts of the function. This formula is essential in finding the intervals where the quadratic function is positive or negative.