Final answer:
To write the equation of a quadratic function with given roots and a y-intercept, factor the quadratic expression using the roots and find the leading coefficient by substituting the y-intercept. The resulting equation is f(x) = (x - 1)(x - 12).
Step-by-step explanation:
The question involves writing an equation of a quadratic function that has roots at (1,0) and (12,0) and a y-intercept at (0,12). Because the equation given is in quadratic form with unknown coefficients, we must find the specific quadratic equation that satisfies these conditions.
A quadratic equation is generally given by ax²+bx+c = 0. With the given points, we know that the quadratic equation we are looking for will pass through these points, which helps us determine the coefficients a, b, and c.
Since the function has roots at x=1 and x=12, we can write the factored form of the quadratic equation as f(x) = a(x - 1)(x - 12). To find the value of a, we can use the y-intercept (0,12). The y-intercept gives us the value of c in the quadratic equation because at the y-intercept x=0.
We know that f(0)=12
in this case. Substituting x=0 into the factored form of the equation, we find that 12 = a(0 - 1)(0 - 12) = 12a.
Dividing both sides by 12, we find that a = 1.
Thus, the equation of the quadratic that passes through the points (1,0), (12,0) with a y-intercept of (0,12) is f(x) = (x - 1)(x - 12).