A rigorous Algebra II lesson investigating key characteristics of parabolas, vertex and standard forms, transformation mechanics, and graphing strategies.
Students complete the 2-page practice set analyzing quadratic anatomy, performing graph transformations, and evaluating a real-world modeling task. Teacher circulates with observation checklist targeting sign errors.
10 Min Wrap-Up
Synthesis, Debrief & Exit Ticket
Review worksheet problem #5 as a whole class. Quick 3-minute exit ticket: "Given \(y = 3(x+4)^2 - 5\), state vertex, direction, and whether it has a maximum or minimum." Collect for formative grading.
Critical Student Misconceptions & Fixes
Misconception: Inside Horizontal Shift Sign
Students see \((x - 4)^2\) and shift 4 units left because of the negative sign.
Correction: Teach setting the inside equal to zero: \(x - 4 = 0 \implies x = +4\).
Misconception: Stretch vs Compression with Signs
Students see \(a = -3\) and think it is a compression because \(-3 < 1\).
Correction: Separate the sign (reflection) from the absolute value \(|a| = 3\) (vertical stretch).
Misconception: Confusing Vertex with \(y\)-Intercept
Assuming \(k\) in vertex form is also where the curve touches the \(y\)-axis.
Correction: Remind students \(y\)-intercept always requires evaluating at \(x = 0\): \(f(0) = a(-h)^2 + k\).
Misconception: Range Interval Endpoints
Writing \((-\infty, k)\) with open parenthesis or confusing \(x\) and \(y\) in range.
Correction: Vertex is reached and included, so always use brackets \([k, \infty)\) or \((-\infty, k]\).
Tiered Differentiation Matrix
Support / Intervention
• Provide coordinate grid templates with labeled \((h, k)\) color tags.
• Limit coefficient \(a\) to \(\pm 1, \pm 2\).
• Use physical tracing paper over parent graph.
On-Grade Mastery
• Complete full 2-page worksheet independently.
• Convert standard form to vertex form using \(h = -b/(2a)\).
• Model real-world trajectory coordinates.
Extension / Honors
• Derive vertex form by completing the square with arbitrary \(a, b, c\).
• Explore horizontal dilations \(y = (bx)^2\) vs vertical \(a\).
Verify that all students calculate symmetry lines before plotting auxiliary points. Students who finish early should complete the real-world flight modeling challenge on Page 2 of the worksheet. Homework: Practice Set 3.2 (Odd problems).
Algebra II Teacher Planning Guide • Unit 3: Quadratics Page 2 of 2
x y
4 Part IV: Real-World Application • Projectile Trajectory (6 Points)
Apply quadratic vertex properties to physical motion.
A soccer ball kicked downfield follows the parabolic trajectory modeled by the equation:
\(h(t) = -16(t - 1.5)^2 + 36\)
where \(h(t)\) represents the height of the ball above the turf in feet, and \(t\) is time in seconds after the kick.
A. Maximum Altitude
What is the maximum height reached by the ball, and at what exact time does it occur?
B. Initial Height
Find the height of the ball at the exact instant it is kicked (\(t = 0\)). Show work.
C. Flight Duration
How many seconds after being kicked does the ball strike the turf (\(h(t) = 0\))?
Algebra II • Curve Detective Worksheet Page 2 of 2
Worked Solutions: Graphing & Real-World Task
Answer Key • Page 2
3 Part III: Graphing via Transformations Key (6 Points)
\(p(x) = -(x - 2)^2 + 4\)
Three Required Transformations (1 pt each):
Horizontal Shift: Shift 2 units right (\(h = 2\))
Reflection: Reflection across \(x\)-axis (\(a = -1\))
Vertical Shift: Shift 4 units up (\(k = 4\))
Completed Symmetric Table (2 pts):
\(x\)
0
1
2 (Vertex)
3
4
\(p(x)\)
0
3
4
3
0
Scoring Breakdown: 1 pt for correctly plotting vertex \((2, 4)\); 1 pt for plotting symmetric pairs \((1, 3), (3, 3)\) and \((0, 0), (4, 0)\); 1 pt for smooth inverted curve with dashed symmetry line \(x = 2\).
