You have the following information about a quadratic function:
- The function passes through point (4,3)
- x-intercepts are -1 and 5
In order to determine which is the quadratic function, you take into account that general form of a quadratic function is:
f(x) = a(x - x1)(x - x2)
where x1 and x2 are the x-intercepts of f(x), given by:
x1 = -1
x2 = 5
By replacing the values of the x-intercepts you obtain:
f(x) = a(x - (-1))(x - 5)
f(x) = a(x + 1)(x - 5)
To find the value of a, you use the fact that the function passes trough point (4,3), that is, for x=4 anf f(x) = 3, you obtain:
3 = a(4 + 1)(4 - 5)
3 = a(5)(-1)
3 = -5a
-3/5 = a
Then, by replacing the value of a into the expression for f(x) you have:
f(x) = -3/5(x + 1)(x - 5)