Algebra 2 Quiz: Graph Polynomial Functions And End Behavior
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Graph Polynomial Functions And End BehaviorQuestion 1 of 20

The polynomial g(x)=x5−3x4−4x3+12x2g(x) = x^5 - 3x^4 - 4x^3 + 12x^2 can be factored as g(x)=x2(x−2)2(x+2)g(x) = x^2(x-2)^2(x+2). Based on this factorization, which characteristic of the graph is most unusual or distinctive?

The graph has exactly three x-intercepts despite being a degree 5 polynomial with complex behavior near each zero
The graph exhibits standard end behavior for a degree 5 polynomial but has an unexpected local maximum between its zeros
The graph touches the x-axis at two different points but never actually crosses the x-axis anywhere
The graph crosses the x-axis only once despite having multiple real zeros due to repeated root effects
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Algebra 2 Quiz

Algebra 2 Quiz: Graph Polynomial Functions And End Behavior

Practice Graph Polynomial Functions And End Behavior in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Graph Polynomial Functions And End Behavior, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The polynomial g(x)=x5−3x4−4x3+12x2g(x) = x^5 - 3x^4 - 4x^3 + 12x^2 can be factored as g(x)=x2(x−2)2(x+2)g(x) = x^2(x-2)^2(x+2). Based on this factorization, which characteristic of the graph is most unusual or distinctive?

  1. The graph has exactly three x-intercepts despite being a degree 5 polynomial with complex behavior near each zero
  2. The graph exhibits standard end behavior for a degree 5 polynomial but has an unexpected local maximum between its zeros
  3. The graph touches the x-axis at two different points but never actually crosses the x-axis anywhere (correct answer)
  4. The graph crosses the x-axis only once despite having multiple real zeros due to repeated root effects
Explanation: When analyzing polynomial graphs, the key insight is understanding how the multiplicity of each zero affects the graph's behavior at that point. The factorization g(x)=x2(x−2)2(x+2)g(x) = x^2(x-2)^2(x+2) reveals three distinct zeros: x=0x = 0 (multiplicity 2), x=2x = 2 (multiplicity 2), and x=−2x = -2 (multiplicity 1). The multiplicity determines whether the graph crosses or touches the x-axis. Odd multiplicities cause the graph to cross through the x-axis, while even multiplicities cause the graph to touch the x-axis and bounce back without crossing. Since zeros at x=0x = 0 and x=2x = 2 both have even multiplicity (2), the graph touches but doesn't cross at these points. Only at x=−2x = -2, with odd multiplicity (1), does the graph actually cross the x-axis. Choice C correctly identifies this unusual behavior: the graph touches the x-axis at two points but crosses only once. Choice A is wrong because having three x-intercepts for a degree 5 polynomial isn't unusual—polynomials can have fewer real zeros than their degree. Choice B incorrectly focuses on end behavior and local maxima, which aren't the distinctive feature here. Choice D reverses the actual situation—the graph crosses once, not "only once despite multiple real zeros." Remember this pattern: when you see repeated factors in a polynomial, immediately check the multiplicities. Even multiplicities create "bounce points" where the graph touches but doesn't cross, while odd multiplicities create crossing points. This visual behavior is often tested on algebra exams.

Question 2

A polynomial function p(x)p(x) has the property that p(x)=x2(x2−9)(2x−6)p(x) = x^2(x^2 - 9)(2x - 6). When analyzing this function's complete behavior, which statement identifies the most significant graphical feature that distinguishes it from simpler polynomial functions?

  1. The function has four x-intercepts with mixed multiplicities, creating a complex pattern of crossing and touching behaviors across its domain
  2. The function has exactly five real zeros when counted with multiplicity, creating more direction changes than typical polynomials
  3. The function demonstrates unusual end behavior due to its mixed positive and negative factors throughout the factored expression
  4. The function has three distinct x-intercepts, with the zero at the origin having multiplicity 2 affecting local curvature significantly (correct answer)
Explanation: When analyzing polynomial functions in factored form, you need to identify zeros and their multiplicities to understand the graph's behavior. Let's examine p(x)=x2(x2−9)(2x−6)p(x) = x^2(x^2 - 9)(2x - 6) systematically. First, find the zeros by setting each factor equal to zero. From x2=0x^2 = 0, you get x=0x = 0 with multiplicity 2. From x2−9=0x^2 - 9 = 0, you get x=±3x = \pm 3 each with multiplicity 1. From 2x−6=02x - 6 = 0, you get x=3x = 3 with multiplicity 1. Since x=3x = 3 appears in two factors, its total multiplicity is 2. This gives you exactly three distinct x-intercepts: x=0x = 0 (multiplicity 2), x=−3x = -3 (multiplicity 1), and x=3x = 3 (multiplicity 2). The even multiplicities at x=0x = 0 and x=3x = 3 mean the graph touches but doesn't cross the x-axis at these points, creating significant local curvature changes. Answer D correctly identifies this pattern. Answer A is wrong because there are only three distinct x-intercepts, not four. Answer B incorrectly counts five zeros with multiplicity—the actual count is four (0, 0, -3, 3). Answer C misunderstands end behavior; since this is a degree-5 polynomial with positive leading coefficient, it follows standard end behavior (down on left, up on right). Remember: when counting zeros in factored form, look for distinct values first, then add up multiplicities from all factors containing each zero. Even multiplicities create "bounce" behavior at intercepts.

Question 3

A polynomial function t(x)t(x) is defined such that its factored form is t(x)=−2(x+4)(x−1)4(x−3)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?

  1. The high multiplicity at x=1x = 1 combined with the negative leading coefficient creates complex local behavior that appears to contradict standard end behavior
  2. The function's degree and leading coefficient create end behavior that conflicts with the typical behavior expected from the pattern of zeros
  3. The even multiplicity at x=1x = 1 creates a flattening effect that must be coordinated with the downward end behavior on both sides (correct answer)
  4. The negative scaling factor affects both the y-intercept value and the function's behavior at each zero location in unexpected ways
Explanation: The function has degree 1+4+1=61 + 4 + 1 = 6 (even) with negative leading coefficient −2-2, so both ends go to −∞-\infty. At x=1x = 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.

Question 4

Consider the polynomial function f(x)=−2x4+8x3−6x2f(x) = -2x^4 + 8x^3 - 6x^2. After factoring completely, which statement correctly describes both the zeros and the end behavior of this function?

  1. Zeros at x=0x = 0 (multiplicity 2), x=1x = 1, and x=3x = 3; as x→±∞x \to \pm\infty, f(x)→−∞f(x) \to -\infty (correct answer)
  2. Zeros at x=0x = 0 (multiplicity 2), x=1x = 1, and x=3x = 3; as x→−∞x \to -\infty, f(x)→+∞f(x) \to +\infty and as x→+∞x \to +\infty, f(x)→−∞f(x) \to -\infty
  3. Zeros at x=0x = 0, x=1x = 1 (multiplicity 2), and x=3x = 3; as x→±∞x \to \pm\infty, f(x)→−∞f(x) \to -\infty
  4. Zeros at x=0x = 0, x=2x = 2, and x=3x = 3; as x→−∞x \to -\infty, f(x)→+∞f(x) \to +\infty and as x→+∞x \to +\infty, f(x)→−∞f(x) \to -\infty
Explanation: First, factor: f(x)=−2x2(x2−4x+3)=−2x2(x−1)(x−3)f(x) = -2x^2(x^2 - 4x + 3) = -2x^2(x-1)(x-3). The zeros are x=0x = 0 (multiplicity 2), x=1x = 1, and x=3x = 3. Since the leading coefficient is negative (-2) and the degree is even (4), as x→±∞x \to \pm\infty, f(x)→−∞f(x) \to -\infty. Choice B has correct zeros but wrong end behavior (mixed instead of both negative). Choice C has wrong multiplicity for x=1x = 1. Choice D has incorrect zeros.

Question 5

A polynomial function has zeros at x=−2x = -2 (multiplicity 1), x=1x = 1 (multiplicity 3), and x=4x = 4 (multiplicity 2). If the function has a negative leading coefficient, which statement about the graph's behavior is correct?

  1. The graph crosses the x-axis at x=−2x = -2 and x=1x = 1, touches at x=4x = 4, and rises to the left while falling to the right
  2. The graph crosses the x-axis at x=−2x = -2 and x=1x = 1, touches at x=4x = 4, and falls to both the left and right (correct answer)
  3. The graph touches the x-axis at x=−2x = -2, crosses at x=1x = 1 and x=4x = 4, and falls to both the left and right
  4. The graph crosses the x-axis at x=−2x = -2, touches at x=1x = 1 and x=4x = 4, and rises to the left while falling to the right
Explanation: Odd multiplicities (1 and 3) mean the graph crosses the x-axis, so it crosses at x=−2x = -2 and x=1x = 1. Even multiplicity (2) means it touches at x=4x = 4. The degree is 1+3+2=61 + 3 + 2 = 6 (even) with negative leading coefficient, so the end behavior is down on both sides. Choice A has wrong end behavior. Choice C confuses crossing/touching behavior. Choice D incorrectly says the graph touches at x=1x = 1 (odd multiplicity should cross).

Question 6

Consider the function h(x)=3x6−12x4+12x2h(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?

  1. The function has three distinct real zeros, and its even degree with positive leading coefficient creates symmetrical end behavior upward
  2. The function has exactly two distinct real zeros with specific multiplicities, and the y-intercept equals the constant term of zero (correct answer)
  3. The function factors into exactly four linear terms, and its graph demonstrates typical polynomial crossing behavior at each zero location
  4. The function exhibits odd symmetry about the origin, and its factored form reveals three real zeros with identical multiplicities throughout
Explanation: Factoring: h(x)=3x2(x4−4x2+4)=3x2(x2−2)2=3x2(x−2)2(x+2)2h(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=0x = 0 (multiplicity 2) and x=±2x = \pm\sqrt{2} (each multiplicity 2), so two distinct values. The y-intercept is h(0)=0h(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.

Question 7

Consider the polynomial s(x)=x6−7x4+12x2s(x) = x^6 - 7x^4 + 12x^2. When this function is graphed, what is the most distinctive characteristic that differentiates its appearance from most other degree-6 polynomials?

  1. The graph exhibits perfect symmetry about the y-axis due to containing only even-powered terms in its expression (correct answer)
  2. The graph has exactly three x-intercepts with alternating crossing and touching behavior creating a distinctive wave pattern
  3. The graph passes through the origin and has two additional x-intercepts that create exactly four turning points along its curve
  4. The graph demonstrates unusual end behavior for a degree-6 polynomial due to its specific coefficient pattern and factored structure
Explanation: Since s(x)=x6−7x4+12x2s(x) = x^6 - 7x^4 + 12x^2 contains only even powers of xx, we have s(−x)=(−x)6−7(−x)4+12(−x)2=x6−7x4+12x2=s(x)s(-x) = (-x)^6 - 7(-x)^4 + 12(-x)^2 = x^6 - 7x^4 + 12x^2 = s(x), making it an even function with y-axis symmetry. Factoring: s(x)=x2(x4−7x2+12)=x2(x2−3)(x2−4)=x2(x−3)(x+3)(x−2)(x+2)s(x) = x^2(x^4 - 7x^2 + 12) = x^2(x^2 - 3)(x^2 - 4) = x^2(x - \sqrt{3})(x + \sqrt{3})(x - 2)(x + 2). Choice B is wrong about alternating behavior (x=0x = 0 has multiplicity 2, so touches). Choice C is wrong about turning points. Choice D is incorrect as the end behavior is standard for degree-6 with positive leading coefficient.

Question 8

Graph p(x)=x(x−2)2(x+1)p(x)=x(x-2)^2(x+1) showing all zeros, indicating which zeros the graph touches or crosses, and giving the end behavior.

  1. Zeros: x=0x=0 (crosses), x=2x=2 (crosses), x=−1x=-1 (crosses); end behavior: as x→±∞x\to\pm\infty, p(x)→∞p(x)\to\infty.
  2. Zeros: x=0x=0 (touches), x=2x=2 (crosses), x=−1x=-1 (crosses); end behavior: as x→±∞x\to\pm\infty, p(x)→∞p(x)\to\infty.
  3. Zeros: x=0x=0 (crosses), x=2x=2 (touches), x=−1x=-1 (crosses); end behavior: as x→−∞x\to-\infty, p(x)→∞p(x)\to\infty and as x→∞x\to\infty, p(x)→−∞p(x)\to-\infty.
  4. Zeros: x=0x=0 (crosses), x=2x=2 (touches), x=−1x=-1 (crosses); end behavior: as x→±∞x\to\pm\infty, p(x)→∞p(x)\to\infty. (correct answer)
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. 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 p(x) = x(x-2)^2(x+1), zeros are x = 0 (mult 1, crosses), x = 2 (mult 2, touches), x = -1 (mult 1, crosses); degree 4 even positive x^4, so both to +∞. Choice A correctly shows these zeros with behaviors and both-up end. Choice B swaps touching and crossing, but recall even multiplicity touches—nice try, you'll get it next time! 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!

Question 9

Graph p(x)=−(x+2)(x−1)2(x−4)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.​

  1. Zeros: x=−2x=-2 (cross), x=1x=1 (touch), x=4x=4 (cross). End behavior: both ends up.
  2. Zeros: x=−2x=-2 (touch), x=1x=1 (cross), x=4x=4 (touch). End behavior: both ends down.
  3. Zeros: x=−2x=-2 (cross), x=1x=1 (touch), x=4x=4 (cross). End behavior: both ends down. (correct answer)
  4. Zeros: x=2x=2 (cross), x=1x=1 (touch), x=4x=4 (cross). End behavior: both ends down.
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. 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. 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.

Question 10

Describe the end behavior of the polynomial h(x)=−3x5+2x3−7x+1h(x)= -3x^5+2x^3-7x+1 as x→±∞x\to\pm\infty.​

  1. As x→−∞x\to-\infty, h(x)→−∞h(x)\to-\infty; as x→+∞x\to+\infty, h(x)→+∞h(x)\to+\infty.
  2. As x→−∞x\to-\infty, h(x)→+∞h(x)\to+\infty; as x→+∞x\to+\infty, h(x)→−∞h(x)\to-\infty. (correct answer)
  3. As x→−∞x\to-\infty, h(x)→+∞h(x)\to+\infty; as x→+∞x\to+\infty, h(x)→+∞h(x)\to+\infty.
  4. As x→−∞x\to-\infty, h(x)→−∞h(x)\to-\infty; as x→+∞x\to+\infty, h(x)→−∞h(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 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.

Question 11

Sketch g(x)=x(x−2)2(x+3)g(x)=x(x-2)^2(x+3) showing zeros, multiplicity behavior (cross/touch), the yy-intercept, and end behavior.

  1. Zeros: x=0x=0 (cross), x=2x=2 (touch), x=−3x=-3 (cross). yy-intercept: (0,0)(0,0). End behavior: left up, right up. (correct answer)
  2. Zeros: x=0x=0 (touch), x=2x=2 (cross), x=−3x=-3 (cross). yy-intercept: (0,0)(0,0). End behavior: left up, right up.
  3. Zeros: x=0x=0 (cross), x=2x=2 (touch), x=−3x=-3 (cross). yy-intercept: (0,0)(0,0). End behavior: left down, right up.
  4. Zeros: x=0x=0 (cross), x=2x=2 (touch), x=−3x=-3 (cross). yy-intercept: (0,6)(0,6). End behavior: left down, right up.
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. 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. For g(x) = x(x-2)^2(x+3), zeros are x = 0 (cross), x = 2 (touch), x = -3 (cross), y-intercept is (0,0), and degree 4 even positive means left up, right up. Choice A correctly shows the zeros with proper cross/touch, y-intercept, and both ends up. Choice D fails by incorrectly using odd positive end behavior (left down, right up) instead of even positive. 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.

Question 12

Graph p(x)=−(x+2)(x−1)2(x−4)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.

  1. Zeros: x=−2x=-2 (cross), x=1x=1 (touch), x=4x=4 (cross). End behavior: both ends down. (correct answer)
  2. Zeros: x=−2x=-2 (cross), x=1x=1 (touch), x=4x=4 (cross). End behavior: both ends up.
  3. Zeros: x=2x=2 (cross), x=1x=1 (touch), x=4x=4 (cross). End behavior: both ends down.
  4. Zeros: x=−2x=-2 (touch), x=1x=1 (cross), x=4x=4 (touch). End behavior: both ends down.
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. End behavior depends ONLY on the leading term axnax^n, because for large ∣x∣|x|, this term dominates all others: in p(x)=2x4−100x3+500x−1000p(x) = 2x^4 - 100x^3 + 500x - 1000, for x=1000x = 1000, the 2x42x^4 term equals 2 trillion while other terms are relatively tiny. For p(x)=−(x+2)(x−1)2(x−4)p(x) = -(x+2)(x-1)^2(x-4), zeros are x=−2x = -2 (cross), x=1x = 1 (touch), x=4x = 4 (cross), and leading term negative x4x^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 nn—count highest power, (2) find sign of leading coefficient aa—look at coefficient of xnx^n term, (3) apply pattern: even nn = both ends match (up if a>0a > 0, down if a<0a < 0); odd nn = ends opposite (if a>0a > 0: ↓↑, if a<0a < 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)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.

Question 13

A polynomial function has been factored as r(x)=(x+1)3(x−2)(x−5)2r(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?

  1. The student counted the number of distinct zeros instead of the total degree when determining end behavior direction (correct answer)
  2. The student assumed that having both odd and even multiplicities creates mixed end behavior on different sides
  3. The student incorrectly determined the leading coefficient sign by focusing on the constant terms within each factor
  4. The student confused the multiplicity effects on zero behavior with their effects on overall end behavior patterns
Explanation: The function r(x)=(x+1)3(x−2)(x−5)2r(x) = (x + 1)^3(x - 2)(x - 5)^2 has degree 3+1+2=63 + 1 + 2 = 6 (even) with positive leading coefficient, so both ends go to +∞+\infty. A common error is counting distinct zeros (3 zeros: x=−1,2,5x = -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.

Question 14

Factor and graph the polynomial function p(x)=x3−x2−6xp(x)=x^3-x^2-6x showing zeros, the yy-intercept, and end behavior. (A rough sketch is sufficient.)

  1. Factored form: x(x−3)(x+2)x(x-3)(x+2); zeros at x=0,3,−2x=0,3,-2 (all cross); yy-intercept (0,0)(0,0); end behavior: left down, right up. (correct answer)
  2. Factored form: x(x−2)(x+3)x(x-2)(x+3); zeros at x=0,2,−3x=0,2,-3; yy-intercept (0,0)(0,0); end behavior: left down, right up.
  3. Factored form: x(x−3)(x+2)x(x-3)(x+2); zeros at x=0,3,−2x=0,3,-2; yy-intercept (0,0)(0,0); end behavior: left up, right down.
  4. Factored form: x(x−3)(x+2)x(x-3)(x+2); zeros at x=0,3,−2x=0,3,-2; yy-intercept (0,6)(0,6); end behavior: left down, right up.
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)=−2x3p(x) = -2x^3 has degree 3 (odd) and leading coefficient -2 (negative), so as x→−∞x \to -\infty, p(x)→+∞p(x) \to +\infty (left end up), and as x→+∞x \to +\infty, p(x)→−∞p(x) \to -\infty (right end down). For p(x)=x3−x2−6x=x(x−3)(x+2)p(x) = x^3 - x^2 - 6x = x(x-3)(x+2), the zeros are x=0,3,−2x = 0, 3, -2 (all multiplicity 1, odd, cross); y-intercept is (0,0)(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 nn—count highest power, (2) find sign of leading coefficient aa—look at coefficient of xnx^n term, (3) apply pattern: even nn = both ends match (up if a>0a > 0, down if a<0a < 0); odd nn = ends opposite (if a>0a > 0: ↓↑\downarrow\uparrow, if a<0a < 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)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!

Question 15

Describe the end behavior of the polynomial p(x)=3x5−2x3+7x−1p(x)=3x^5-2x^3+7x-1 as x→±∞x \to \pm \infty.

  1. As x→±∞x \to \pm \infty, p(x)→∞p(x) \to \infty.
  2. As x→−∞x \to -\infty, p(x)→∞p(x) \to \infty and as x→∞x \to \infty, p(x)→−∞p(x) \to -\infty.
  3. As x→±∞x \to \pm \infty, p(x)→−∞p(x) \to -\infty.
  4. As x→−∞x \to -\infty, p(x)→−∞p(x) \to -\infty and as x→∞x \to \infty, p(x)→∞p(x) \to \infty. (correct answer)
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. End behavior depends ONLY on the leading term axnax^n, because for large ∣x∣|x|, this term dominates all others: in p(x)=2x4−100x3+500x−1000p(x) = 2x^4 - 100x^3 + 500x - 1000, for x=1000x = 1000, the 2x42x^4 term equals 2 trillion while other terms are relatively tiny. The four end behavior patterns are: (1) even degree + positive aa = both ends up, (2) even degree + negative aa = both ends down, (3) odd degree + positive aa = left down, right up, (4) odd degree + negative aa = left up, right down. Memorize these four! For p(x)=3x5−2x3+7x−1p(x) = 3x^5 - 2x^3 + 7x - 1, the leading term is 3x53x^5 (degree 5 odd, positive), so as x→−∞x \to -\infty, p(x)→−∞p(x) \to -\infty and as x→∞x \to \infty, p(x)→∞p(x) \to \infty. Choice B correctly describes this left-down, right-up behavior for odd positive leading. A choice like A reverses it, but remember the pattern for positive odd: left down, right up—you're getting the hang of it! End behavior shortcut: (1) find degree nn—count highest power, (2) find sign of leading coefficient aa—look at coefficient of xnx^n term, (3) apply pattern: even nn = both ends match (up if a>0a > 0, down if a<0a < 0); odd nn = ends opposite (if a>0a > 0: ↓↑, if a<0a < 0: ↑↓). Example: −3x5+100x2−50-3x^5 + 100x^2 - 50 has degree 5 (odd), a=−3a = -3 (negative), so left up, right down. Ignore all other terms—only the leading term matters for end behavior!

Question 16

Graph p(x)=2(x−1)2(x+2)3p(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.)​

  1. Zeros: x=1x=1 (touches), x=−2x=-2 (crosses with flattening); end behavior: left down, right up. (correct answer)
  2. Zeros: x=1x=1 (crosses), x=−2x=-2 (touches); end behavior: left down, right up.
  3. Zeros: x=1x=1 (touches), x=−2x=-2 (crosses with flattening); end behavior: left up, right down.
  4. Zeros: x=1x=1 (touches), x=2x=2 (crosses with flattening); end behavior: left down, right up.
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). 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!

Question 17

For the polynomial p(x)=−x3+6x2−9x,p(x)=-x^3+6x^2-9x, factor to find the zeros and state the end behavior (enough to sketch).

  1. p(x)=−x(x−3)2p(x)=-x(x-3)^2; zeros: x=0x=0 (mult. 1), x=3x=3 (mult. 2); end behavior: as x→−∞x\to-\infty, p(x)→∞p(x)\to\infty and as x→∞x\to\infty, p(x)→−∞p(x)\to-\infty. (correct answer)
  2. p(x)=−x(x−3)2p(x)=-x(x-3)^2; zeros: x=0x=0 (mult. 1), x=3x=3 (mult. 2); end behavior: as x→±∞x\to\pm\infty, p(x)→−∞p(x)\to-\infty.
  3. p(x)=−x(x−3)2p(x)=-x(x-3)^2; zeros: x=0x=0 (mult. 2), x=3x=3 (mult. 1); end behavior: as x→−∞x\to-\infty, p(x)→−∞p(x)\to-\infty and as x→∞x\to\infty, p(x)→∞p(x)\to\infty.
  4. p(x)=−x(x−3)(x+3)p(x)=-x(x-3)(x+3); zeros: x=0,3,−3x=0,3,-3; end behavior: as x→−∞x\to-\infty, p(x)→∞p(x)\to\infty and as x→∞x\to\infty, p(x)→−∞p(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. 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 p(x) = -x^3 + 6x^2 - 9x = -x(x-3)^2, zeros x = 0 (mult 1), x = 3 (mult 2); degree 3 odd negative, so left +∞, right -∞. Choice A correctly factors, gives multiplicities, and matches the end behavior. Choice C swaps multiplicities and reverses end behavior, but odd negative is left up right down—practice the patterns! 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!

Question 18

Graph p(x)=x5−5x3+4xp(x)=x^5-5x^3+4x by finding its zeros (from factoring) and using end behavior to make a rough sketch.

  1. Zeros: x=0,±1,±2x=0,\pm1,\pm2 (all cross). End behavior: left down, right up. (correct answer)
  2. Zeros: x=0,±1,±2x=0,\pm1,\pm2 (all touch). End behavior: left down, right up.
  3. Zeros: x=0,±1,±2x=0,\pm1,\pm2 (all cross). End behavior: left up, right down.
  4. Zeros: x=0,±1x=0,\pm1 only (all cross). End behavior: left down, right up.
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). Factoring p(x)=x5−5x3+4xp(x) = x^5 - 5x^3 + 4x gives x(x−2)(x+2)(x−1)(x+1)x(x-2)(x+2)(x-1)(x+1), with zeros x=0,±1,±2x = 0, \pm1, \pm2 (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.

Question 19

Graph p(x)=x5−5x3+4xp(x)=x^5-5x^3+4x by finding its zeros (from factoring) and using end behavior to make a rough sketch.​

  1. Zeros: x=0,±1,±2x=0,\pm1,\pm2 (all cross). End behavior: left down, right up. (correct answer)
  2. Zeros: x=0,±1,±2x=0,\pm1,\pm2 (all touch). End behavior: left down, right up.
  3. Zeros: x=0,±1,±2x=0,\pm1,\pm2 (all cross). End behavior: left up, right down.
  4. Zeros: x=0,±1x=0,\pm1 only (all cross). End behavior: left down, right up.
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). 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.

Question 20

For the polynomial q(x)=−x5+4x3−x,q(x) = -x^5 + 4x^3 - x, describe the end behavior as x→±∞x \to \pm \infty.

  1. As x→−∞x \to -\infty, q(x)→−∞q(x) \to -\infty and as x→+∞x \to +\infty, q(x)→+∞q(x) \to +\infty.
  2. As x→±∞x \to \pm \infty, q(x)→+∞q(x) \to +\infty.
  3. As x→−∞x \to -\infty, q(x)→+∞q(x) \to +\infty and as x→+∞x \to +\infty, q(x)→−∞q(x) \to -\infty. (correct answer)
  4. As x→±∞x \to \pm \infty, q(x)→−∞q(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. End behavior depends ONLY on the leading term axnax^n, because for large ∣x∣|x|, this term dominates all others: in p(x)=2x4−100x3+500x−1000p(x) = 2x^4 - 100x^3 + 500x - 1000, for x=1000x = 1000, the 2x42x^4 term equals 2 trillion while other terms are relatively tiny—the four end behavior patterns are: (1) even degree + positive aa = both ends up, (2) even degree + negative aa = both ends down, (3) odd degree + positive aa = left down, right up, (4) odd degree + negative aa = left up, right down—memorize these four! For q(x)=−x5+4x3−xq(x) = -x^5 + 4x^3 - x, the leading term is −x5-x^5 (degree 5 odd, negative coefficient), so as x→−∞x \to -\infty, q(x)→+∞q(x) \to +\infty, and as x→+∞x \to +\infty, q(x)→−∞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 nn—count highest power, (2) find sign of leading coefficient aa—look at coefficient of xnx^n term, (3) apply pattern: even nn = both ends match (up if a>0a > 0, down if a<0a < 0); odd nn = ends opposite (if a>0a > 0: ↓↑, if a<0a < 0: ↑↓)—for example, −3x5+100x2−50-3x^5 + 100x^2 - 50 has degree 5 (odd), a=−3a = -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)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!