Questions: Which equation could possibly represent the graphed function? A. f(x)=(x-4)(x+2)(x+4) B. f(x)=(x-4)^2(x-2) C. f(x)=(x+4)^2(x+2) D. f(x)=(x-4)(x-2)(x+4)

Which equation could possibly represent the graphed function? A. f(x)=(x-4)(x+2)(x+4) B. f(x)=(x-4)^2(x-2) C. f(x)=(x+4)^2(x+2) D. f(x)=(x-4)(x-2)(x+4)
Transcript text: Which equation could possibly represent the graphed function? A. $f(x)=(x-4)(x+2)(x+4)$ B. $f(x)=(x-4)^{2}(x-2)$ C. $f(x)=(x+4)^{2}(x+2)$ D. $f(x)=(x-4)(x-2)(x+4)$
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Solution

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Solution Steps

Step 1: Identify the x-intercepts of the graph

The graph intersects the x-axis at \( x = -4 \), \( x = -2 \), and \( x = 4 \).

Step 2: Determine the possible factors of the polynomial

Given the x-intercepts, the factors of the polynomial must be \( (x + 4) \), \( (x + 2) \), and \( (x - 4) \).

Step 3: Match the factors with the given options

Check which option has the factors \( (x + 4) \), \( (x + 2) \), and \( (x - 4) \):

  • Option A: \( f(x) = (x - 4)(x + 2)(x + 4) \)
  • Option B: \( f(x) = (x - 4)^2(x - 2) \)
  • Option C: \( f(x) = (x + 4)^2(x + 2) \)
  • Option D: \( f(x) = (x - 4)(x - 2)(x + 4) \)

Final Answer

The correct equation that represents the graphed function is: \[ \boxed{A. \ f(x) = (x - 4)(x + 2)(x + 4)} \]

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