The function provided is:

$y = \ln\left(-(x - 1)(x - 3)\right)$

To graph this, let’s analyze the expression step-by-step:

Step 1: Analyze the Argument of the Logarithm

The logarithmic function is only defined for positive arguments. Therefore, the expression inside the logarithm must be greater than zero:

$-(x - 1)(x - 3) > 0$

We need to find the intervals where this expression is positive.

Step 2: Solve the Inequality

First, determine the zeros of the expression:

$-(x - 1)(x - 3) = 0$

The zeros are at $x = 1$ and $x = 3$.

Now, analyze the sign of the expression in each interval divided by these points:

1. For $x < 1$: Both $(x - 1)$ and $(x - 3)$ are negative, so $-(x - 1)(x - 3)$ is positive.
2. For $1 < x < 3$: $(x - 1)$ is positive and $(x - 3)$ is negative, so $-(x - 1)(x - 3)$ is negative (undefined for the logarithm).
3. For $x > 3$: Both $(x - 1)$ and $(x - 3)$ are positive, so $-(x - 1)(x - 3)$ is negative (undefined for the logarithm).

Valid Interval for the Function

The logarithm is defined only for $x < 1$.

Step 3: Express the Function in the Valid Interval

For $x < 1$:

$y = \ln\left(-(x - 1)(x - 3)\right)$

This expression simplifies to:

$y = \ln\left((1 - x)(3 - x)\right)$

Step 4: Graph the Function

The function is only defined for $x < 1$. As $x$ approaches 1 from the left, the function tends toward negative infinity because the logarithmic term approaches zero.

The graph will be a curve in the left region of $x < 1$, with a vertical asymptote at $x = 1$ and the function diverging downward.

Would you like me to plot this graph for you?