Algebra 2 Question of the Day

1. $f(x)=(x^2-1)(x^2-4)$; zeros at $x=\pm1,\pm2$; end behavior: as $x\to\pm\infty$, $f(x)\to-\infty$.
2. $f(x)=(x-1)(x-4)(x+1)(x+4)$; zeros at $x=\pm1,\pm4$; end behavior: as $x\to\pm\infty$, $f(x)\to\infty$.
3. $f(x)=(x^2-1)(x^2-4)$; zeros at $x=\pm1,\pm2$ (crosses at each); end behavior: as $x\to\pm\infty$, $f(x)\to\infty$. (correct answer)
4. $f(x)=(x^2+1)(x^2-4)$; zeros at $x=\pm2$ only; end behavior: as $x\to\pm\infty$, $f(x)\to\infty$.

Explanation: This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. Graphing a polynomial requires two main elements: (1) zeros (found from factored form by setting each factor equal to zero) with their multiplicities determining whether the graph crosses (odd multiplicity) or touches (even multiplicity) at each zero, and (2) end behavior (determined by the leading term's degree and sign). For example, p(x) = -2x³ has degree 3 (odd) and leading coefficient -2 (negative), so as x → -∞, p(x) → +∞ (left end up), and as x → +∞, p(x) → -∞ (right end down). These two elements give you the skeleton of the graph! For f(x) = x^4 - 5x^2 + 4 = (x^2-1)(x^2-4) = (x-1)(x+1)(x-2)(x+2), zeros ±1, ±2 (all mult 1, crosses); degree 4 even positive, both to +∞. Choice A correctly factors and shows all zeros crossing with both-up end. Choice D has the right factoring but wrong end (both down), but positive leading means up—keep that sign in mind! The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior. You don't need exact turning points—just show the zeros, their behavior, and where the graph ends up as x → ±∞!