Final answer:
The quadratic function with x-intercepts at -4 and -6 and a y-intercept at 72 is f(x) = 3x² + 30x + 72. This is determined by starting with the factored form f(x) = a(x + 4)(x + 6) and finding the coefficient 'a' using the y-intercept.
Step-by-step explanation:
The question involves finding the equation for a quadratic function that has x-intercepts at x = -4 and x = -6 and a y-intercept at y = 72. To form the equation, we start with the factored form of a quadratic equation that fits the given intercepts, which is:
f(x) = a(x + 4)(x + 6)
Since the y-intercept occurs when x = 0, and the function yields y = 72 at that point, we can substitute x with 0 to find the value of a:
72 = a(0 + 4)(0 + 6)
Solving for a gives us:
a = 72 / 24
a = 3
So the equation of the quadratic function is:
f(x) = 3(x + 4)(x + 6)
Expanding this, we get:
f(x) = 3(x^2 + 10x + 24)
f(x) = 3x^2 + 30x + 72