Graph Polynomial Functions and End Behavior - Algebra 2

1. Factor and sketch $g(x)=x^3-4x^2-x+4$ showing all real zeros and the end behavior.
   Answer: $g(x)=(x-1)(x-4)(x+1)$. Zeros: $x=1,4,-1$. End behavior: left down, right up. 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. End behavior depends ONLY on the leading term $ax^n$, because for large |x|, this term dominates all others: in p(x) = 2x⁴ - 100x³ + 500x - 1000, for x = 1000, the 2x⁴ term equals 2 trillion while other terms are relatively tiny. Factoring g(x) = $x^3$ - $4x^2$ - x + 4 gives (x-1)(x-4)(x+1), with zeros x = 1, 4, -1 (all cross), and degree 3 odd positive means left down, right up. Choice A correctly provides the factored form, zeros, and end behavior. Choice B fails by stating incorrect end behavior (left up, right down) despite the same factors, which would require a negative leading coefficient. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). 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.
2. Graph $p(x)=x^5-5x^3+4x$ by finding its zeros (from factoring) and using end behavior to make a rough sketch.​
   Answer: Zeros: $x=0,\pm1,\pm2$ (all cross). End behavior: left down, right up. 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). Factoring p(x) = $x^5$ - $5x^3$ + 4x gives x(x-2)(x+2)(x-1)(x+1), with zeros x = 0, ±1, ±2 (all cross), and degree 5 odd positive means left down, right up. Choice A correctly identifies the zeros as all crossing and the end behavior. Choice C fails by reversing the end behavior to odd negative, which requires a negative leading coefficient. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). 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.
3. Describe the end behavior of the polynomial $g(x)=5x^4-3x^2+7.$​
   Answer: As $x\to\pm\infty$, $g(x)\to+\infty$. 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. End behavior depends ONLY on the leading term $ax^n$, because for large |x|, this term dominates all others: in p(x) = 2x⁴ - 100x³ + 500x - 1000, for x = 1000, the 2x⁴ term equals 2 trillion while other terms are relatively tiny—the four end behavior patterns are: (1) even degree + positive a = both ends up, (2) even degree + negative a = both ends down, (3) odd degree + positive a = left down, right up, (4) odd degree + negative a = left up, right down—memorize these four! For g(x) = $5x^4$ - $3x^2$ + 7, the leading term is $5x^4$ (degree 4 even, positive coefficient), so as x → ±∞, g(x) → +∞ (both ends up). Choice C correctly identifies the end behavior as both ends approaching positive infinity, aligning with the even positive pattern. A distractor like choice A might confuse it with odd degree behavior, but always ignore lower terms and focus on the highest power. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓)—ignore all other terms—only the leading term matters for end behavior! 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're building strong skills!
4. For the polynomial function $g(x)=x^3-x^2-6x,$ identify the zeros and describe the end behavior (based on the leading term).
   Answer: Zeros: $x=-2,0,3$. End behavior: as $x\to-\infty$, $g(x)\to-\infty$ and as $x\to+\infty$, $g(x)\to+\infty$. 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 g(x) = x³ - x² - 6x, first factor: g(x) = x(x² - x - 6) = x(x-3)(x+2). Setting each factor to zero gives zeros: x=0, x=3, and x=-2. The leading term is x³ with degree 3 (odd) and leading coefficient +1 (positive), so the end behavior follows the odd/positive pattern: as x→-∞, g(x)→-∞ (left down) and as x→+∞, g(x)→+∞ (right up). Choice A correctly identifies zeros at x=-2, 0, 3 and end behavior: as x→-∞, g(x)→-∞ and as x→+∞, g(x)→+∞. Choice C has the correct zeros but incorrect end behavior—with odd degree and positive leading coefficient, the ends must go in opposite directions (left down, right up), not both up! End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). Example: -3x⁵ + 100x² - 50 has degree 5 (odd), a = -3 (negative), so left up, right down. Ignore all other terms—only the leading term matters for end behavior!
5. Describe the end behavior of the polynomial function $f(x)=-3x^5+2x^3-7.$
   Answer: As $x\to-\infty$, $f(x)\to+\infty$ and as $x\to+\infty$, $f(x)\to-\infty$. 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. End behavior depends ONLY on the leading term $ax^n$, because for large |x|, this term dominates all others: in p(x) = 2x⁴ - 100x³ + 500x - 1000, for x = 1000, the 2x⁴ term equals 2 trillion while other terms are relatively tiny. The four end behavior patterns are: (1) even degree + positive a = both ends up, (2) even degree + negative a = both ends down, (3) odd degree + positive a = left down, right up, (4) odd degree + negative a = left up, right down. Memorize these four! For f(x) = -3x⁵ + 2x³ - 7, the leading term is -3x⁵ with degree 5 (odd) and leading coefficient -3 (negative). Since we have odd degree with negative leading coefficient, the pattern is: left end up, right end down. This means as x→-∞, f(x)→+∞ and as x→+∞, f(x)→-∞. Choice B correctly identifies this end behavior: as x→-∞, f(x)→+∞ and as x→+∞, f(x)→-∞. Choice A incorrectly reverses the end behavior—remember that odd degree with negative leading coefficient gives left up, right down, not left down, right up! 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 → ±∞!
6. For the polynomial $q(x) = -x^5 + 4x^3 - x,$ describe the end behavior as $x \to \pm \infty$.
   Answer: As $x \to -\infty$, $q(x) \to +\infty$ and as $x \to +\infty$, $q(x) \to -\infty$. 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. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $p(x) = 2x^4 - 100x^3 + 500x - 1000$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny—the four end behavior patterns are: (1) even degree + positive $a$ = both ends up, (2) even degree + negative $a$ = both ends down, (3) odd degree + positive $a$ = left down, right up, (4) odd degree + negative $a$ = left up, right down—memorize these four! For $q(x) = -x^5 + 4x^3 - x$, the leading term is $-x^5$ (degree 5 odd, negative coefficient), so as $x \to -\infty$, $q(x) \to +\infty$, and as $x \to +\infty$, $q(x) \to -\infty$ (left up, right down). Choice B correctly describes the end behavior for this odd negative degree polynomial. A distractor like choice C might treat it as even degree, but confirm the highest power is odd here. End behavior shortcut: (1) find degree $n$—count highest power, (2) find sign of leading coefficient $a$—look at coefficient of $x^n$ term, (3) apply pattern: even $n$ = both ends match (up if $a > 0$, down if $a < 0$); odd $n$ = ends opposite (if $a > 0$: ↓↑, if $a < 0$: ↑↓)—for example, $-3x^5 + 100x^2 - 50$ has degree 5 (odd), $a = -3$ (negative), so left up, right down—ignore all other terms! 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—keep it up!
7. Consider the function $h(x) = 3x^6 - 12x^4 + 12x^2$. When this function is completely factored and its graph is analyzed, which statement correctly describes a key relationship between its algebraic and graphical properties?
   Answer: The function has exactly two distinct real zeros with specific multiplicities, and the y-intercept equals the constant term of zero Factoring: $h(x) = 3x^2(x^4 - 4x^2 + 4) = 3x^2(x^2 - 2)^2 = 3x^2(x - \sqrt{2})^2(x + \sqrt{2})^2$. This gives exactly two distinct real zeros: $x = 0$ (multiplicity 2) and $x = \pm\sqrt{2}$ (each multiplicity 2), so two distinct values. The y-intercept is $h(0) = 0$, which equals the constant term. Choice A is wrong about three zeros. Choice C is wrong about linear terms and crossing behavior (all multiplicities are even, so touching only). Choice D is wrong about odd symmetry and identical multiplicities.
8. Graph the polynomial function $p(x)=(x+2)^2(x-1)(x-3)$ showing all zeros (with correct crossing/touching behavior), the $y$-intercept, and the end behavior.
   Answer: Zeros: $x=-2$ (touches), $x=1$ (crosses), $x=3$ (crosses); end behavior: as $x\to\pm\infty$, $p(x)\to\infty$; $y$-intercept: $(0,12)$. 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 p(x) = $(x+2)^2$(x-1)(x-3), the zeros are x = -2 (multiplicity 2, touches), x = 1 (multiplicity 1, crosses), and x = 3 (multiplicity 1, crosses); the degree is 4 (even) with positive leading coefficient, so both ends go to +∞; the y-intercept is p(0) = 12. Choice B correctly shows zeros at x = -2 (touches), x = 1 (crosses), x = 3 (crosses), both ends up, and y-intercept (0,12). A common mistake, like in choice A, is swapping the touching and crossing behaviors or misidentifying the end behavior for an even degree polynomial—remember, even degree with positive leading means both ends up! 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 → ±∞!
9. Graph $p(x)=-(x+2)(x-1)^2(x-4)$ showing zeros, whether the graph crosses or touches at each zero, and the correct end behavior.
   Answer: Zeros: $x=-2$ (cross), $x=1$ (touch), $x=4$ (cross). End behavior: both ends down. 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. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $p(x) = 2x^4 - 100x^3 + 500x - 1000$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny. For $p(x) = -(x+2)(x-1)^2(x-4)$, zeros are $x = -2$ (cross), $x = 1$ (touch), $x = 4$ (cross), and leading term negative $x^4$ (even degree, negative), so both ends down. Choice C correctly shows the zeros with proper cross/touch and end behavior both down. Choice A fails by incorrectly stating both ends up, which would be for positive leading coefficient. End behavior shortcut: (1) find degree $n$—count highest power, (2) find sign of leading coefficient $a$—look at coefficient of $x^n$ term, (3) apply pattern: even $n$ = both ends match (up if $a > 0$, down if $a < 0$); odd $n$ = ends opposite (if $a > 0$: ↓↑, if $a < 0$: ↑↓). 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.
10. Graph the polynomial function $p(x)=(x+2)^2(x-1)(x-3)$ showing all zeros (with whether the graph crosses or touches at each zero) and the correct end behavior.
    Answer: Zeros: $x=-2$ (touches), $x=1$ (crosses), $x=3$ (crosses). End behavior: as $x\to-\infty$, $p(x)\to+\infty$ and as $x\to+\infty$, $p(x)\to+\infty$. 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 the polynomial p(x) = $(x+2)^2$(x-1)(x-3), the zeros are x = -2 (multiplicity 2, touches), x = 1 (multiplicity 1, crosses), and x = 3 (multiplicity 1, crosses), and the degree is 4 (even) with a positive leading coefficient, so both ends go to +∞. Choice A correctly shows the zeros with proper touch/cross behavior and the end behavior matching the even degree and positive coefficient. A common distractor like Choice D fails by incorrectly stating the end behavior as left down and right up, which would be for an odd degree with positive coefficient instead of even. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). 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.
11. Graph $p(x)=2(x-1)^2(x+2)^3$ showing zeros with multiplicities, whether the graph crosses or touches at each zero, and the correct end behavior. (Rough sketch.)
    Answer: Zeros: $x=1$ (touches), $x=-2$ (crosses with flattening); end behavior: left down, right up. 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). For p(x) = 2$(x-1)^2$$(x+2)^3$, the zeros are x = 1 (multiplicity 2, even, touches), x = -2 (multiplicity 3, odd, crosses with flattening); degree 5 (odd) with positive leading coefficient $2x^5$, so left down, right up. Choice A correctly shows the zeros with touching/crossing (noting flattening for multiplicity 3) and the end behavior for odd positive degree. A distractor like choice C might reverse the end behavior, but positive odd degrees always go down left and up right. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓)—for example, -3x⁵ + 100x² - 50 has degree 5 (odd), a = -3 (negative), so left up, right down. 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—fantastic effort!
12. A polynomial function $t(x)$ is defined such that its factored form is $t(x) = -2(x + 4)(x - 1)^4(x - 3)$. When sketching this function's graph, which combination of features requires the most careful attention to avoid common graphing errors?
    Answer: The even multiplicity at $x = 1$ creates a flattening effect that must be coordinated with the downward end behavior on both sides The function has degree $1 + 4 + 1 = 6$ (even) with negative leading coefficient $-2$, so both ends go to $-\infty$. At $x = 1$, the multiplicity 4 (even) creates a very flat touching behavior where the graph barely grazes the x-axis. Students often struggle to draw this flattening effect correctly while maintaining the overall downward end behavior. Choice A misunderstands that local and end behaviors don't contradict. Choice B incorrectly suggests conflict between degree/coefficient and zero pattern. Choice D overstates the scaling factor's complexity.
13. Graph $f(x)=x(x-4)(x+1)^2$ showing all zeros, whether the graph crosses or touches at each zero, and the end behavior.
    Answer: Zeros: $x=0$ (crosses), $x=4$ (crosses), $x=-1$ (touches). End behavior: as $x\to-\infty$, $f(x)\to+\infty$ and as $x\to+\infty$, $f(x)\to+\infty$. 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(x-4)(x+1)², we find zeros: x=0 (multiplicity 1, odd, so crosses), x-4=0 gives x=4 (multiplicity 1, odd, so crosses), and x+1=0 gives x=-1 (multiplicity 2, even, so touches). The degree is 1+1+2=4 (even) and the leading coefficient is +1 (positive when expanded), so both ends go up: as x→-∞, f(x)→+∞ and as x→+∞, f(x)→+∞. Choice A correctly identifies zeros at x=0 (crosses), x=4 (crosses), x=-1 (touches) with end behavior: as x→-∞, f(x)→+∞ and as x→+∞, f(x)→+∞. Choice C has the correct zeros but incorrect end behavior—with even degree and positive leading coefficient, both ends must go up, not have the left end going down! 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 → ±∞!
14. Factor and graph the polynomial function $p(x)=x^3-x^2-6x$ showing zeros, the $y$-intercept, and end behavior. (A rough sketch is sufficient.)
    Answer: Factored form: $x(x-3)(x+2)$; zeros at $x=0,3,-2$ (all cross); $y$-intercept $(0,0)$; end behavior: left down, right up. 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) = -2x^3$ has degree 3 (odd) and leading coefficient -2 (negative), so as $x \to -\infty$, $p(x) \to +\infty$ (left end up), and as $x \to +\infty$, $p(x) \to -\infty$ (right end down). For $p(x) = x^3 - x^2 - 6x = x(x-3)(x+2)$, the zeros are $x = 0, 3, -2$ (all multiplicity 1, odd, cross); y-intercept is $(0,0)$; degree 3 (odd) with positive leading coefficient, so left down, right up. Choice A correctly provides the factored form, zeros with crossing, y-intercept, and end behavior matching the odd positive degree. A distractor like choice C might flip the end behavior, but remember odd degrees always have opposite ends, with the sign deciding which way. End behavior shortcut: (1) find degree $n$—count highest power, (2) find sign of leading coefficient $a$—look at coefficient of $x^n$ term, (3) apply pattern: even $n$ = both ends match (up if $a > 0$, down if $a < 0$); odd $n$ = ends opposite (if $a > 0$: $\downarrow\uparrow$, if $a < 0$: $\uparrow\downarrow$). 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—great job factoring first!
15. Sketch $g(x)=-(x+3)(x-2)^2(x-4)$ showing where it crosses or touches the $x$-axis and the end behavior.
    Answer: Zeros: $x=-3$ (crosses), $x=2$ (touches), $x=4$ (crosses); end behavior: as $x\to\pm\infty$, $g(x)\to-\infty$. 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. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $p(x) = 2x^4 - 100x^3 + 500x - 1000$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny. The four end behavior patterns are: (1) even degree + positive $a$ = both ends up, (2) even degree + negative $a$ = both ends down, (3) odd degree + positive $a$ = left down, right up, (4) odd degree + negative $a$ = left up, right down. Memorize these four! For $g(x) = -(x+3)(x-2)^2(x-4)$, zeros are $x = -3$ (mult 1, crosses), $x = 2$ (mult 2, touches), $x = 4$ (mult 1, crosses); degree 4 even with negative leading $-x^4$, so both to $-\infty$. Choice A correctly identifies the zeros with proper behaviors and the both-down end behavior. In choice C, the end behavior is listed as both up, but that's for positive leading coefficient—remember to check the sign! 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 \to \pm\infty$!
16. Describe the end behavior of the polynomial $h(x)= -3x^5+2x^3-7x+1$ as $x\to\pm\infty$.​
    Answer: As $x\to-\infty$, $h(x)\to+\infty$; as $x\to+\infty$, $h(x)\to-\infty$. 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 h(x) = -$3x^5$ + $2x^3$ - 7x + 1, the leading term is -$3x^5$ (odd degree, negative coefficient), so as x → -∞, h(x) → +∞ and as x → +∞, h(x) → -∞. Choice B correctly describes this end behavior for odd degree negative. A distractor like Choice A reverses it to odd positive, which would require a positive leading coefficient. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). 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.
17. For $q(x)=2(x-3)(x+1)(x+2),$ identify the zeros and state the end behavior.
    Answer: Zeros: $x=3,-1,-2$. End behavior: as $x\to-\infty$, $q(x)\to-\infty$ and as $x\to+\infty$, $q(x)\to+\infty$. 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 q(x) = 2(x-3)(x+1)(x+2), we find zeros by setting each factor to zero: x-3=0 gives x=3, x+1=0 gives x=-1, and x+2=0 gives x=-2. All have multiplicity 1 (odd), so the graph crosses at each zero. The degree is 1+1+1=3 (odd) and the leading coefficient is +2 (positive), so we have the odd/positive pattern: as x→-∞, q(x)→-∞ (left down) and as x→+∞, q(x)→+∞ (right up). Choice A correctly identifies zeros at x=3, -1, -2 with end behavior: as x→-∞, q(x)→-∞ and as x→+∞, q(x)→+∞. Choice B has the correct zeros but incorrect end behavior—with odd degree and positive leading coefficient, we get left down, right up, not left up, right down! End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). Example: -3x⁵ + 100x² - 50 has degree 5 (odd), a = -3 (negative), so left up, right down. Ignore all other terms—only the leading term matters for end behavior!
18. Sketch the polynomial $f(x)=(x-2)(x+1)(x+4)$ identifying all zeros and the end behavior. Which description matches the correct sketch?
    Answer: Zeros at $x=-4,-1,2$ (all cross); end behavior: left down, right up. 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). For f(x) = (x-2)(x+1)(x+4), the zeros are x = 2, -1, -4 (all multiplicity 1, odd, cross); degree 3 (odd) with positive leading coefficient $x^3$, so left down, right up. Choice A correctly lists the zeros (noting x = -4, -1, 2) with crossing and the end behavior for odd positive degree. A distractor like choice B might reverse the end behavior, but recall that positive odd degrees go down on left and up on right. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). 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're mastering this!
19. Graph $p(x)=2(x-1)^2(x+2)^3$ showing zeros with multiplicities, whether the graph crosses or touches at each zero, and the correct end behavior. (Rough sketch.)​
    Answer: Zeros: $x=1$ (touches), $x=-2$ (crosses with flattening); end behavior: left down, right up. 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). For p(x) = 2$(x-1)^2$$(x+2)^3$, the zeros are x = 1 (multiplicity 2, even, touches), x = -2 (multiplicity 3, odd, crosses with flattening); degree 5 (odd) with positive leading coefficient $2x^5$, so left down, right up. Choice A correctly shows the zeros with touching/crossing (noting flattening for multiplicity 3) and the end behavior for odd positive degree. A distractor like choice C might reverse the end behavior, but positive odd degrees always go down left and up right. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓)—for example, -3x⁵ + 100x² - 50 has degree 5 (odd), a = -3 (negative), so left up, right down. 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—fantastic effort!
20. A polynomial function has been factored as $r(x) = (x + 1)^3(x - 2)(x - 5)^2$. If a student incorrectly identifies the end behavior of this function, which misconception most likely led to their error?
    Answer: The student counted the number of distinct zeros instead of the total degree when determining end behavior direction The function $r(x) = (x + 1)^3(x - 2)(x - 5)^2$ has degree $3 + 1 + 2 = 6$ (even) with positive leading coefficient, so both ends go to $+\infty$. A common error is counting distinct zeros (3 zeros: $x = -1, 2, 5$) instead of total degree (6) for end behavior. Students might think odd number of zeros means odd degree. Choice B is incorrect because multiplicities don't affect end behavior. Choice C is wrong because the leading coefficient comes from the highest degree terms. Choice D misunderstands that multiplicities affect local behavior at zeros, not global end behavior.

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