Question

identify the equation for the graph.

• \\(y = (x-1)^3(x-2)^2(x-4)^3\\)
• \\(y = (x-1)(x-2)^2(x-4)^3\\)
• \\(y = (x-1)^3(x-2)^2(x-4)\\)
• \\(y = (x-1)(x-2)^2(x-4)\\)

Explanation:

Identify the roots from the graph

The graph intersects or touches the x-axis at three distinct points:

$$ x = 1, \quad x = 2, \quad x = 4 $$

Determine multiplicity at each root

Using the Polynomial Multiplicity and Graph Behavior at Roots knowledge points

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Match with the given options

We analyze the exponents of the factors \( (x-1) \), \( (x-2) \), and \( (x-4) \) in each option:

• Option 1: \( y = (x-1)^3(x-2)^2(x-4)^3 \)
• Multiplicities: \( x=1 \) is 3 (crosses), \( x=2 \) is 2 (touches), \( x=4 \) is 3 (crosses). Total degree is 8.
• Option 2: \( y = (x-1)(x-2)^2(x-4)^3 \)
• Multiplicities: \( x=1 \) is 1 (crosses), \( x=2 \) is 2 (touches), \( x=4 \) is 3 (crosses). Total degree is 6.
• Option 3: \( y = (x-1)^3(x-2)^2(x-4) \)
• Multiplicities: \( x=1 \) is 3 (crosses), \( x=2 \) is 2 (touches), \( x=4 \) is 1 (crosses). Total degree is 6.
• Option 4: \( y = (x-1)(x-2)^2(x-4) \)
• Multiplicities: \( x=1 \) is 1 (crosses), \( x=2 \) is 2 (touches), \( x=4 \) is 1 (crosses). Total degree is 4.

Analyze end behavior and inflection

We observe the behavior at the crossings:

• At \( x = 1 \), the graph crosses very steeply and directly, resembling a linear crossing (multiplicity 1).
• At \( x = 4 \), the graph flattens out slightly as it crosses, indicating an inflection point characteristic of a cubic crossing (multiplicity 3).
• Both ends of the graph point upwards as \( x \to \pm\infty \), which requires an even overall degree.
• Checking the degrees:
• Option 2 has degree \( 1 + 2 + 3 = 6 \) (even degree, correct end behavior).
• Option 4 has degree \( 1 + 2 + 1 = 4 \) (even degree, but lacks the cubic flattening at \( x = 4 \)).
• Therefore, the factor \( (x-4) \) must have a multiplicity of 3, and \( (x-1) \) has a multiplicity of 1. This matches \( y = (x-1)(x-2)^2(x-4)^3 \).

Answer:

• y = (x-1)^3(x-2)^2(x-4)^3
• y = (x-1)(x-2)^2(x-4)^3 (Correct answer)
• y = (x-1)^3(x-2)^2(x-4)
• y = (x-1)(x-2)^2(x-4)