To find the equation of a quadratic function with given zeros and a point, you can start by using the zero-factor theorem. If the zeros are -1 and 3, then the factors of the quadratic equation are (x + 1) and (x - 3).
Now, we can write the equation in factored form:
f(x) = a(x + 1)(x - 3)
To determine the value of 'a,' we can use the given point (1, 4):
f(1) = 4
Substituting this into the equation:
4 = a(1 + 1)(1 - 3)
4 = a(2)(-2)
Now, solve for 'a':
4 = -4a
Divide both sides by -4:
a = -1
So, the equation of the quadratic function in factored form is:
f(x) = -1(x + 1)(x - 3)
You can leave it in this factored form or expand it if needed:
f(x) = -1(x^2 - 2x - 3)
f(x) = -x^2 + 2x + 3
This is the equation of the quadratic function with the given zeros (-1 and 3) and the point (1, 4).