Question

write an equation for the polynomial graphed below

\\(y(x) = \\)

Explanation:

⚡ Using what you learned: Graphing Polynomial Functions

Step 1: Identify the roots and their multiplicities

Observe the \(x\)-intercepts (roots) of the graph and how the curve behaves at each intercept:

• At \(x = -2\), the graph crosses the \(x\)-axis directly. This indicates a single root with multiplicity 1: factor is \((x + 2)\).
• At \(x = -1\), the graph crosses the \(x\)-axis directly. This indicates a single root with multiplicity 1: factor is \((x + 1)\).
• At \(x = 1\), the graph touches the \(x\)-axis and turns around (tangent to the axis). This indicates a root with an even multiplicity, typically 2: factor is \((x - 1)^2\).
• At \(x = 4\), the graph crosses the \(x\)-axis directly. This indicates a single root with multiplicity 1: factor is \((x - 4)\).

The general form of the polynomial is:

$$ y(x) = a(x + 2)(x + 1)(x - 1)^2(x - 4) $$

Step 2: Find the leading coefficient \(a\)

Use the \(y\)-intercept from the graph to solve for the constant \(a\).
The graph crosses the \(y\)-axis at \((0, -4)\).

Substitute \(x = 0\) and \(y(0) = -4\) into the equation:

$$ -4 = a(0 + 2)(0 + 1)(0 - 1)^2(0 - 4) $$

$$ -4 = a(2)(1)(1)(-4) $$

$$ -4 = -8a $$

$$ a = \frac{-4}{-8} = 0.5 $$

Step 3: Write the final equation

Substitute \(a = 0.5\) back into the polynomial equation:

$$ y(x) = 0.5(x + 2)(x + 1)(x - 1)^2(x - 4) $$

Answer:

$$ y(x) = 0.5(x + 2)(x + 1)(x - 1)^2(x - 4) $$