Question

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Explanation:

Identify the roots and their multiplicities from the graph

The graph crosses the \(x\)-axis at \(x = -2\) (linear crossing, multiplicity 1).
The graph crosses the \(x\)-axis at \(x = 2\) (linear crossing, multiplicity 1).
The graph touches and turns around at the \(x\)-axis at \(x = 3\) (tangent, multiplicity 2).
This gives the general factored form:

$$y = a(x + 2)(x - 2)(x - 3)^2$$

Determine the leading coefficient using the y-intercept

The graph passes through the \(y\)-intercept at \((0, -36)\).
Substitute \(x = 0\) and \(y = -36\) into the equation:

$$-36 = a(0 + 2)(0 - 2)(0 - 3)^2$$

$$-36 = a(2)(-2)(9)$$

$$-36 = -36a \implies a = 1$$

Match with the given options

The equation is:

$$y = (x + 2)(x - 2)(x - 3)^2$$

This matches the option:

$$y = (x + 2)(x - 2)(x - 3)^2$$

Answer:

• (A) \(y = (x + 2)(x - 2)(x - 3)\)
• (B) \(y = (x + 2)(x - 2)(x - 3)^2\) (Correct answer)
• (C) \(y = (x + 2)(x - 2)^2(x - 3)^2\)
• (D) \(y = (x + 2)^2(x - 2)(x - 3)^2\)