write a function in any form that would match the graph shown below.\n$f(x) = $

Explanation:

Step1: Identify the x-intercepts and their multiplicities

The graph touches the x-axis at $x = -5$ (multiplicity 2) and crosses at $x = 6$ (multiplicity 1).

Step2: Write the general polynomial form

$$f(x) = a(x + 5)^2(x - 6)$$

Step3: Use the y-intercept to find the constant $a$

The graph passes through $(0, -100)$. Substitute $x = 0$ and $f(x) = -100$. $$-100 = a(0 + 5)^2(0 - 6)$$

Step4: Solve for the leading coefficient $a$

$$-100 = a(25)(-6) \implies -100 = -150a \implies a = \frac{-100}{-150} = \frac{2}{3}$$

Step5: Construct the final function

$$f(x) = \frac{2}{3}(x + 5)^2(x - 6)$$

Answer:

f(x) = \frac{2}{3}(x + 5)^2(x - 6)