Final answer:
To write the quadratic function with given roots (3, 0) and (1, 0) and a point (4, 9), we start with the factored form y = a(x - 3)(x - 1), substitute the point to find 'a', and then expand to obtain y = 3x^2 - 12x + 9.
Step-by-step explanation:
The student has asked how to write a quadratic function given the roots (3, 0) and (1, 0), and a point (4, 9). Since the roots are known, the quadratic function can be written in factored form as
y = a(x - 3)(x - 1),
where 'a' is a constant that we need to determine. By using the given point (4, 9), we can substitute the x and y values into the equation to solve for 'a':
9 = a(4 - 3)(4 - 1),
9 = a(1)(3),
9 = 3a,
a = 3.
Now we have the quadratic function:
y = 3(x - 3)(x - 1).
This can be expanded to get the standard form of the quadratic equation:
y = 3(x^2 - 4x + 3),
y = 3x^2 - 12x + 9.