Sketch the graph of the function 𝑓(𝑥) = −(𝑥 − 4)(𝑥 + 1)2(𝑥 − 5). Explain your thinking.

To sketch the graph of the function 𝑓(𝑥) = −(𝑥 − 4)(𝑥 + 1)²(𝑥 − 5), we need to follow these steps:

1. Identify the roots of the function: The roots of the function are the values of x that make the function equal to zero. In this case, the roots are x = 4, x = -1, and x = 5.

2. Determine the multiplicity of the roots: The multiplicity of a root is determined by the power of the factor in the function. In this case, the root x = -1 has multiplicity 2 (because of the square), and the roots x = 4 and x = 5 each have multiplicity 1.

3. Determine the end behavior of the function: The end behavior of the function is determined by the leading term, which is -x^4 in this case. This means that as x approaches positive or negative infinity, the function approaches negative infinity.

4. Plot the roots and sketch the function: Plot the roots on the x-axis and draw the function starting from the bottom left (since the function approaches negative infinity as x approaches negative infinity), going through each root according to its multiplicity (touching the x-axis at roots with even multiplicity and crossing the x-axis at roots with odd multiplicity), and ending at the bottom right (since the function approaches negative infinity as x approaches positive infinity).

5. Check the behavior around the roots: The function should be decreasing on the intervals (-∞, -1), (-1, 4), and (5, ∞) and increasing on the interval (4, 5) because of the negative sign in front of the function.

Remember, this is a rough sketch and may not capture all the nuances of the function's behavior. For a more accurate graph, you could use graphing software or a graphing calculator.