Leading Term: _________ Degree: _____
\( \lim_{x \to \infty} f(x) = \) ________ \( \lim_{x \to -\infty} f(x) = \) ________
Leading Term & Degree Work:
Work:
Root 1: \( x = 0 \) (Mult: ___) Behavior: _________
Root 2: \( x = -5 \) (Mult: ___) Behavior: _________
Root 3: \( x = 2 \) (Mult: ___) Behavior: _________
Leading Term: _________ Degree: _____ (Even/Odd)
Leading Term & Multiplicity Analysis:
Work:
Root at 0: Mult ___ Behavior: _____________
Root at 2: Mult ___ Behavior: _____________
Root at -1: Mult ___ Behavior: _____________
End Behavior: As \( x \to \pm\infty \), \( h(x) \to \) ________
Rewrite \( (2 - x)^3 \) in terms of \( (x - 2)^3 \):
Work:
Root at 2: Mult ___ Behavior: _____________
Root at -3: Mult ___ Behavior: _____________
Leading Term: _________ Sign: _____ (+ / -)
\( \lim_{x \to \infty} j(x) = \) ________ \( \lim_{x \to -\infty} j(x) = \) ________
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Name
Seat #
Date
ALGEBRA II Education is Freedom
U02R02 – SWBAT Analyze root multiplicity, predict graph behavior at intercepts, and model quadratic functions
Practice Set A – Core Task 1 (Graph to Factored Form)
Target Function Graph: x y -1 2 1 2 -2
[ ] (A) \( p(x) = x^2(2 - x)^3(x + 1) \)
[ ] (B) \( p(x) = x^2(x - 2)^3(x + 1) \)
[ ] (C) \( p(x) = x(2 - x)^3(x + 1)^2 \)
[ ] (D) \( p(x) = x(2 - x)^3(x - 1) \)
Everybody Writes:
Explain step-by-step why Option (B) is correct and why Option (A) must be eliminated. Explicitly address: 1) The behavior at each root (\( x = 0 \), \( x = 2 \), \( x = -1 \)) and their corresponding multiplicities. 2) How the leading term and end behavior prove whether \( (x - 2)^3 \) or \( (2 - x)^3 \) must be used.
Write your complete explanation here...
Problem Set A – Additional Practice (Root Behaviors)
At \( x = -4 \):
Mult: _____ Behavior: _________
At \( x = 1 \):
Mult: _____ Behavior: _________
At \( x = 3 \):
Mult: _____ Behavior: _________
\( f(x) = \)
© 2026 Uncommon Schools, Inc. All rights reserved. Page 3 of 5
Name
Seat #
Date
ALGEBRA II Education is Freedom
U02R02 – SWBAT Analyze root multiplicity, predict graph behavior at intercepts, and model quadratic functions
Problem Set B – Additional Practice (Quadratic Forms & Formulas)
[ ] (A) The parabola opens downward.
[ ] (B) The parabola is the same width as \( y = x^2 \).
[ ] (C) The maximum value of the function is 16.
[ ] (D) The vertex is at \( (-2, 16) \).
Justify: Explain why the selected statement is false and state the correct vertex coordinates:
Work / Explanation:
(A)
(B)
(C)
(D)
Show Factoring or Vieta's \( xy = \frac{c}{a} \):
Value of \( -xy = \) _____________
Show algebraic steps to isolate \( l \):
\( l = \) ___________________________
© 2026 Uncommon Schools, Inc. All rights reserved. Page 4 of 5
Name
Seat #
Date
ALGEBRA II Education is Freedom
U02R02 – SWBAT Analyze root multiplicity, predict graph behavior at intercepts, and model quadratic functions
EXIT TICKET (UNIT 2 QUIZ SIMULATION)
Notice: crosses at \( x = -2 \) and flattens/inflects at \( x = 1 \).
[ ] (A) \( f(x) = (x + 2)(x - 1)^3 \)
[ ] (B) \( f(x) = -(x + 2)(x - 1)^3 \)
[ ] (C) \( f(x) = -(x - 2)(x + 1)^3 \)
[ ] (D) \( f(x) = (x + 2)^2(x - 1)^2 \)
State the limit notation for:
\( g(x) = -2(x - 5)^2(3 - x)^3 \)
Leading Term Work:
Leading Term: _________
\( \lim_{x \to \infty} g(x) = \) __________
\( \lim_{x \to -\infty} g(x) = \) __________
For \( f(x) = 2(x + 3)^2 - 8 \):
a) Vertex: ( _____ , _____ )
b) Opens: Up / Down
c) Min Value: _________
d) Width vs \( y = x^2 \):
[ ] Narrower [ ] Wider
Vertex Form Mastery
Find \( -xy \) where \( x \) and \( y \) are roots to:
\( 2t^2 - 7t - 6 = 0 \)
Work / Vieta's:
\( -xy = \) _____________
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Algebra II • Unit 2 Quiz Prep Teacher Answer Key Page 1 of 2
Day 1 Key Continued: Practice Sets A & B + Exit Ticket Teacher Resource
Packet Page 3: Practice Set A (Classifying & Everybody Writes)
• 1a) \( f(x) = x^4 - 2 \implies \) (B) Quartic Binomial
• 1b) \( g(x) = 3x^2 + 2x + 1 \implies \) (D) Quadratic Trinomial
• 1c) \( h(x) = 9 - x^3 \implies \) (C) Cubic Binomial
• 1d) \( j(x) = 4 \implies \) (A) Constant Monomial
• a) \( -5x^4 + 3x + 7 \implies \) Quartic Trinomial (Degree 4, 3 terms)
• b) \( -18 \implies \) Constant Monomial (Degree 0, 1 term)
• c) \( (3x-1)(x+4) = 3x^2+11x-4 \implies \) Quadratic Trinomial
Exemplar "Everybody Writes" Student Response:
"To classify \( h(x) = 9 - x^3 \), I must first write it in standard form by ordering terms from greatest to least exponent: \( h(x) = -x^3 + 9 \). The highest exponent is 3, making it cubic, and there are 2 terms, making it a binomial. Standard form is essential because students who look only at the first term might mistake 9 as the degree or leading coefficient, leading to the incorrect classification of a constant monomial."
Packet Page 4: Practice Set B (Algebraic Proofs, Tables & SAT Style)
• a) \( f(x) = 12x^{11} - 6x^7 - 5x^3 \):
\( f(-x) = -12x^{11} + 6x^7 + 5x^3 = -f(x) \implies \) ODD
• b) \( g(x) = 9x^8 - x^4 + 2x^2 \):
\( g(-x) = 9x^8 - x^4 + 2x^2 = g(x) \implies \) EVEN
• Table: \( f(3) = 24.0 \implies f(-3) = \mathbf{24.0} \); \( f(-5) = 10.5 \implies f(5) = \mathbf{10.5} \).
• Point \( (-4, 9) \): Even \(\implies \mathbf{(4, 9)}\); Odd \(\implies \mathbf{(4, -9)}\).
\( E = \frac{O + 4M + P}{6} \implies 6E = O + 4M + P \implies \mathbf{P = 6E - O - 4M}\). Answer: (A)
\( 5t^2 + 8t - 4 = 0 \). Product \( xy = \frac{c}{a} = \frac{-4}{5} \). Thus, \( -xy = -(-\frac{4}{5}) = \mathbf{\frac{4}{5}} \). Answer: (B)
Packet Page 5: EXIT TICKET Answer Key & Scoring Guide (4 Points Total) 1 pt each
Q1: Odd Graph
Answer: (A)
Origin symmetry (\( 180^\circ \) rotation maps onto itself; passes through \( (0,0) \)).
Q2: End Behavior
Answer: (C)
Leading term \( -4x^7 \). As \( x \to \infty, y \to -\infty \); as \( x \to -\infty, y \to \infty \).
Q3: Even Point
Point: (-2, -5)
Reflects across \( y \)-axis: \( (x, y) \to (-x, y) \). \( x \) changes sign, \( y \) stays same.
Q4: Odd Point
Point: (-2, 5)
Rotates \( 180^\circ \) about origin: \( (x, y) \to (-x, -y) \). Both coordinates flip signs.
Algebra II • Unit 2 Quiz Prep Teacher Answer Key Page 2 of 2
Algebra II • Unit 2 Quiz Prep Teacher Answer Key Page 1 of 2
Day 2 Key Continued: Practice Sets A & B + Exit Ticket Teacher Resource
Packet Page 3: Practice Set A (Quiz Q4 Deconstruction & Everybody Writes)
Correct Answer: (B) \( p(x) = x^2(x - 2)^3(x + 1) \)
Eliminate (C) because \( (x+1)^2 \) would mean a bounce at \( -1 \), but graph crosses. Eliminate (D) because root is \( +1 \) instead of \( -1 \).
• At \( x = -4 \): Mult 1 \(\implies\) Crosses
• At \( x = 1 \): Mult 2 \(\implies\) Bounces
• At \( x = 3 \): Mult 3 \(\implies\) Wiggles / Inflects
• 3) \( f(x) = -x^2(x+3)(x-5) \) (Degree 4, opens down)
Exemplar "Everybody Writes" Student Response:
"Option (B) is the correct function because it matches both the intercept behaviors and the end behavior. At \( x = -1 \), the graph cuts straight through the axis, which requires a factor of \( (x + 1)^1 \). At \( x = 0 \), the graph touches the origin and turns around, indicating an even multiplicity factor of \( x^2 \). At \( x = 2 \), the graph flattens horizontally as it passes through, indicating an odd multiplicity factor of \( (x - 2)^3 \). Furthermore, both arrows point upward as \( x \to \pm\infty \), which requires a positive leading coefficient. In Option (A), the factor \( (2 - x)^3 = -(x - 2)^3 \) introduces a negative leading term (\( -x^6 \)), forcing both ends to point downward. Therefore, Option (A) is eliminated and Option (B) is correct."
Packet Page 4: Practice Set B (Quadratic Features & Quiz Boosters)
Answer: (D) is NOT TRUE
\( f(x) = -(x - 2)^2 + 16 \). In vertex form \( a(x - h)^2 + k \), the vertex is \( (h, k) = \mathbf{(2, 16)} \), NOT \( (-2, 16) \). The negative sign inside \( (x - 2) \) means \( h = +2 \).
Answer: (A)
\( f(x) = -(x + 2)(x - 1) \). \( x \)-intercepts are \( x = -2 \) and \( x = 1 \). Negative leading coefficient means opens down. \( y \)-intercept is \( -(2)(-1) = +2 \). Matches Graph (A).
\( 3t^2 + 10t - 8 = 0 \). Product of roots \( xy = \frac{c}{a} = \frac{-8}{3} \). Therefore, \( -xy = -(-\frac{8}{3}) = \mathbf{\frac{8}{3}} \).
\( A = \pi r^2 + \pi r l \implies A - \pi r^2 = \pi r l \implies \mathbf{l = \frac{A - \pi r^2}{\pi r}} \) or \( \mathbf{l = \frac{A}{\pi r} - r} \).
Packet Page 5: EXIT TICKET Answer Key & Scoring Guide (4 Points Total) 1 pt each
Q1: Polynomial Graph
Answer: (B)
Crosses at \( -2 \) \(\implies (x+2)^1\); wiggles at \( 1 \) \(\implies (x-1)^3\); negative lead coeff \(\implies -(x+2)(x-1)^3\).
Q2: End Behavior
Leading: \( +2x^5 \)
\( (-2)(x^2)(-x^3) = 2x^5 \). \( \lim_{x \to \infty} = \mathbf{\infty} \), \( \lim_{x \to -\infty} = \mathbf{-\infty} \).
Q3: Vertex Form
Vertex: (-3, -8)
Opens Up; Min = -8; Narrower than \( y = x^2 \) because \( |a| = 2 > 1 \).
Q4: Quadratic Roots
Value: 3
\( 2t^2 - 7t - 6 = 0 \). \( xy = \frac{c}{a} = \frac{-6}{2} = -3 \). Thus, \( -xy = -(-3) = \mathbf{3} \).
Algebra II • Unit 2 Quiz Prep Teacher Answer Key Page 2 of 2
