Question

identify the equation for the graph. $y = (x - 1)(x - 3)^2(x - 5)$ $y = (x - 1)^3(x - 3)(x - 5)^3$ $y = (x - 1)^2(x - 3)(x - 5)^2$ $y = (x - 1)(x - 3)^3(x - 5)$

Explanation:

Step1: Analyze x-intercepts and multiplicities

The graph intersects the x - axis at \(x = 1\), \(x=3\), and \(x = 5\). At \(x = 3\), the graph touches the x - axis and turns around, which means the multiplicity of the root \(x = 3\) is even. At \(x=1\) and \(x = 5\), the graph crosses the x - axis, so the multiplicities of \(x = 1\) and \(x=5\) are odd.

Step2: Analyze the options

• Option 1: \(y=(x - 1)(x - 3)^{2}(x - 5)\). The multiplicity of \(x = 1\) is 1 (odd), \(x = 3\) is 2 (even), \(x=5\) is 1 (odd). This matches the behavior of the graph (crosses at \(x = 1\) and \(x = 5\), touches at \(x=3\)).
• Option 2: \(y=(x - 1)^{3}(x - 3)(x - 5)^{3}\). Multiplicities of \(x = 1\) (3, odd), \(x=3\) (1, odd), \(x = 5\) (3, odd). The graph would cross at all three points, which does not match.
• Option 3: \(y=(x - 1)^{2}(x - 3)(x - 5)^{2}\). Multiplicities of \(x = 1\) (2, even), \(x=3\) (1, odd), \(x = 5\) (2, even). The graph would touch at \(x = 1\) and \(x=5\), cross at \(x = 3\), which does not match.
• Option 4: \(y=(x - 1)(x - 3)^{3}(x - 5)\). Multiplicity of \(x = 3\) is 3 (odd), so the graph would cross at \(x = 3\), which does not match the touch - and - turn behavior at \(x = 3\).

Answer:

\(y=(x - 1)(x - 3)^{2}(x - 5)\) (the first option)