help get the graphical representation for this  y = ln (-(x - 1)*(x - 3))

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.

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Explore how to graph the logarithmic function y = ln(-(x - 1)(x - 3)) step-by-step. Understand the domain restrictions and graph behavior around x = 1 and x = 3. Suitable for advanced high school students and those studying calculus.

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