The graph of a polynomial function is shown in the xy-plane. Which of the following could be an equation for...

1. TRANSLATE the graph features into mathematical information

From the graph, identify where the curve crosses the x-axis:

• X-intercepts (zeros): \(\mathrm{x = -3}\), \(\mathrm{x = 1}\), and \(\mathrm{x = 6}\)

2. INFER the multiplicity at each zero

This is the crucial step! Look at HOW the graph behaves at each x-intercept:

• At \(\mathrm{x = -3}\): The curve crosses straight through the x-axis
  • This indicates odd multiplicity (most likely 1)
• At \(\mathrm{x = 1}\): The curve touches the x-axis but turns around without crossing
  • This indicates even multiplicity (most likely 2)
  • Think of it like a ball bouncing off the x-axis
• At \(\mathrm{x = 6}\): The curve crosses straight through the x-axis
  • This indicates odd multiplicity (most likely 1)

3. INFER the sign of the leading coefficient

Observe the end behavior:

• As x goes far left (negative), y goes down (negative)
• As x goes far right (positive), y also goes down (negative)
• Both ends pointing downward means negative leading coefficient

4. TRANSLATE zeros into factored form

• A zero at \(\mathrm{x = -3}\) means a factor of \(\mathrm{(x + 3)}\)
• A zero at \(\mathrm{x = 1}\) with multiplicity 2 means a factor of \(\mathrm{(x - 1)^2}\)
• A zero at \(\mathrm{x = 6}\) means a factor of \(\mathrm{(x - 6)}\)

So we're looking for: \(\mathrm{y = (negative\,constant)(x + 3)(x - 1)^2(x - 6)}\)

5. APPLY CONSTRAINTS to eliminate wrong answers

• Eliminate C and D immediately: They have zeros at \(\mathrm{x = 3}\), \(\mathrm{x = -1}\), and \(\mathrm{x = -6}\), which don't match our graph
• Compare A and B:
  • Choice A: \(\mathrm{y = -\frac{1}{8}(x + 3)(x - 1)(x - 6)}\) → multiplicity 1 at all zeros
  • Choice B: \(\mathrm{y = -\frac{1}{8}(x + 3)(x - 1)^2(x - 6)}\) → multiplicity 2 at \(\mathrm{x = 1}\)
• Since the graph clearly shows touching behavior at \(\mathrm{x = 1}\), we need multiplicity 2

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the difference between crossing and touching behavior at x-intercepts

Many students correctly identify the three x-intercepts but don't carefully analyze HOW the graph interacts with the x-axis at each point. They see that the curve hits \(\mathrm{x = -3}\), \(\mathrm{1}\), and \(\mathrm{6}\), and immediately look for any equation with those three zeros, leading them to select Choice A without considering multiplicities.

The key distinction is subtle but critical: at \(\mathrm{x = 1}\), the graph doesn't cross through—it bounces off the x-axis like a ball hitting the ground. This "bounce" is the visual signature of even multiplicity.

Second Most Common Error:

Weak TRANSLATE skill: Confusing the signs in factored form

Students sometimes mix up whether a zero at \(\mathrm{x = 1}\) corresponds to a factor of \(\mathrm{(x - 1)}\) or \(\mathrm{(x + 1)}\). If they incorrectly translate \(\mathrm{x = 1}\) as requiring \(\mathrm{(x + 1)}\), or \(\mathrm{x = -3}\) as requiring \(\mathrm{(x - 3)}\), they might be drawn toward Choice C or D even though these have completely different x-intercepts visible on the graph.

The Bottom Line:

This problem tests whether you can "read" a polynomial graph by connecting visual features (crossing vs. touching) to algebraic properties (multiplicity). The x-intercepts tell you WHAT the zeros are, but the BEHAVIOR at each intercept tells you the multiplicity—and that's what distinguishes the correct answer.