Question

5.5 roots and zeros
example 4
write a polynomial that could
41.

Explanation:

Step1: Identify x-intercepts (roots)

From the graph, the polynomial crosses the x-axis at $x=-2$ and $x=2$. These are the roots, so the factors are $(x+2)$ and $(x-2)$.

Step2: Form basic polynomial

Multiply the factors: $(x+2)(x-2) = x^2 - 4$.

Step3: Account for vertical stretch/compression

The vertex of the parabola is at $(0, -4)$. Substitute $x=0$ into $y=a(x^2 - 4)$:
$-4 = a(0^2 - 4)$
$-4 = -4a$
Solve for $a$: $a = 1$.

Answer:

$y = x^2 - 4$