Question

drag the tiles to the correct boxes. not all tiles will be used. which equation could represent each graphed polynomial function? $y = (x + 1)(x - 3)(x^2 + 1)$ $y = x^3 + 27$ $y = x^4 - 5x^2 + 4$

Explanation:

Step1: Identify x-intercepts from graph

The graph crosses the x-axis at $x=-2$ and $x=2$, so the function has factors of $(x+2)$ and $(x-2)$.

Step2: Analyze end behavior

The graph opens upwards on both ends, so the leading coefficient is positive, and the degree is even.

Step3: Test candidate equations

1. For $y=(x+2)(x-2)(x^2+1)$:

Expand the first two factors: $(x^2-4)(x^2+1) = x^4 + x^2 -4x^2 -4 = x^4 -3x^2 -4$. This has x-intercepts at $x=\pm2$, but its y-intercept is $-4$, which does not match the graph's y-intercept of $4$.

1. For $y=x^4 -5x^2 +4$:

Factor the polynomial: $x^4 -5x^2 +4=(x^2-1)(x^2-4)=(x-1)(x+1)(x-2)(x+2)$. The x-intercepts are $x=\pm1, \pm2$ (matches the graph's touches/crosses at these points). The y-intercept is $4$ (substitute $x=0$: $0 -0 +4=4$), which matches the graph. The leading term is $x^4$, so end behavior matches the upward-opening even-degree graph.

1. For $y=x^4+2x$:

Substitute $x=0$: $y=0$, which does not match the y-intercept of $4$.

Answer:

$y = x^4 - 5x^2 + 4$