Question

identify the equation for the graph.
y = (x - 1)(x - 2)^2(x - 4)^3
y = (x - 1)(x - 2)^2(x - 4)
y = (x - 1)^3(x - 2)^2(x - 4)
y = (x - 1)^3(x - 2)^2(x - 4)^3

Explanation:

Step1: Recall root - multiplicity rules

If \(y=(x - a)^n\), when \(n\) is odd, the graph crosses the \(x\) - axis at \(x = a\), and when \(n\) is even, the graph touches the \(x\) - axis at \(x=a\).

Step2: Analyze the roots of the graph

The graph touches the \(x\) - axis at \(x = 2\) and crosses the \(x\) - axis at \(x=1\) and \(x = 4\). So the factor corresponding to \(x = 2\) should have an even exponent, and the factors corresponding to \(x=1\) and \(x = 4\) should have odd exponents.

Step3: Check the options

For the factor \((x - 1)\), it should have an odd exponent, for \((x - 2)\) it should have an even exponent and for \((x - 4)\) it should have an odd exponent. The equation \(y=(x - 1)(x - 2)^2(x - 4)^3\) satisfies these conditions.

Answer:

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