Question

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

Explanation:

Step1: Write the factored - form of quadratic function

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 the factored - form. We have \(3=a(4 - 1)(4 - 5)\). Simplify the right - hand side: \(3=a\times3\times(-1)\), which is \(3=-3a\). Solving for \(a\), we get \(a=-1\).

Step3: Expand the factored form to standard form

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

Answer:

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