Question

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Explanation:

⚡ Using what you learned: Graphing Polynomial Functions

Step 1: Identify the x-intercepts (roots)

The graph crosses the x-axis at four distinct points:

$$ x = -4, \quad x = -1, \quad x = 1, \quad x = 3 $$

Since the graph crosses directly through the x-axis at each of these points, each root has a multiplicity of 1.

Step 2: Write the factored form

Using the roots, write the polynomial function in its factored form:

$$ f(x) = a(x + 4)(x + 1)(x - 1)(x - 3) $$

Step 3: Solve for the leading coefficient

The graph passes through the y-intercept at \( (0, 3) \). Substitute \( x = 0 \) and \( f(0) = 3 \) into the equation to find \( a \):

$$ 3 = a(0 + 4)(0 + 1)(0 - 1)(0 - 3) $$

$$ 3 = a(4)(1)(-1)(-3) $$

$$ 3 = 12a $$

$$ a = \frac{3}{12} = \frac{1}{4} $$

Step 4: Write the final equation

Substitute \( a = \frac{1}{4} \) back into the factored form:

$$ f(x) = \frac{1}{4}(x + 4)(x + 1)(x - 1)(x - 3) $$

Optionally, simplify the difference of squares \( (x + 1)(x - 1) = x^2 - 1 \):

$$ f(x) = \frac{1}{4}(x + 4)(x^2 - 1)(x - 3) $$

Answer:

$$ f(x) = \frac{1}{4}(x + 4)(x + 1)(x - 1)(x - 3) $$