write a quadratic function in standard form whose graph is shown. the function is f(x) =

Explanation:

Step1: Use factored - form of quadratic

Since the x - intercepts are (x = 1) and (x = 5), the factored form of the quadratic function is (f(x)=a(x - 1)(x - 5)).

Step2: Find the value of (a)

Substitute the point ((4,3)) into (f(x)=a(x - 1)(x - 5)). We get (3=a(4 - 1)(4 - 5)). Simplify the right - hand side: (3=a(3)(-1)), so (3=-3a). Solving for (a), we divide both sides by (-3) and find (a=-1).

Step3: Convert to standard form

Substitute (a = - 1) into (f(x)=a(x - 1)(x - 5)), we have (f(x)=-(x - 1)(x - 5)). Expand ((x - 1)(x - 5)=x^{2}-5x-x + 5=x^{2}-6x + 5). Then (f(x)=-x^{2}+6x - 5).

Answer:

(-x^{2}+6x - 5)