Curve Shifters Worksheet
Algebra II Unit 2 • Functions
Curve Shifters: Function Transformations
Mastery Worksheet
Total Points: / 40
Name:
Date:
Period:
Part 1: Transformation Architecture • \(g(x) = a \cdot f(b(x - h)) + k\)
Coordinate Rule: \((x, y) \to \left(\frac{x}{b} + h,\; a \cdot y + k\right)\)
Parameter \(a\) (Outside)
Vertical stretch \((|a| > 1)\) or shrink \((0 < |a| < 1)\). If \(a < 0\), reflect over \(x\)-axis.
Parameter \(b\) (Inside)
Horizontal shrink \((|b| > 1)\) or stretch \((0 < |b| < 1)\) by \(\frac{1}{|b|}\). If \(b < 0\), reflect over \(y\)-axis.
Parameter \(h\) (Inside)
Horizontal shift: \((x - h)\) moves RIGHT \(h\); \((x + h)\) moves LEFT \(h\).
Parameter \(k\) (Outside)
Vertical shift: \(+k\) moves UP \(k\); \(-k\) moves DOWN \(k\) units.
Part 2: Parameter Extraction & Transformation Sequences
Identify the parent function \(f(x)\) and list all transformations in operational order.
1. \(g(x) = -3(x - 4)^2 + 7\) [4 pts]
Parent Function:
Reflection(s):
Stretch / Shrink:
Translations:
2. \(g(x) = \frac{1}{2}\sqrt{x + 5} - 6\) [4 pts]
Parent Function:
Reflection(s):
Stretch / Shrink:
Translations:
3. \(g(x) = 4|-(x - 1)| - 8\) [4 pts]
Parent Function:
Reflection(s):
Stretch / Shrink:
Translations:
4. \(g(x) = -\left(\frac{1}{3}x\right)^3 + 2\) [4 pts]
Parent Function:
Reflection(s):
Stretch / Shrink:
Translations:
Part 3: Coordinate Mapping Rules
Apply the mapping formula to transform specific coordinate anchor points.
- For \(f(x) = x^2\), parent point is \((3, 9)\).
\(g(x) = 2(x - 5)^2 - 3\)
Mapping Formula: \((x, y) \to ( \quad\quad\quad , \quad\quad\quad )\)
New Point \((\quad\quad , \quad\quad)\)
- For \(f(x) = \sqrt{x}\), parent point is \((4, 2)\).
\(g(x) = -\sqrt{x + 1} + 5\)
Mapping Formula: \((x, y) \to ( \quad\quad\quad , \quad\quad\quad )\)
New Point \((\quad\quad , \quad\quad)\)
- For \(f(x) = |x|\), parent point is \((-4, 4)\).
\(g(x) = \frac{1}{4}|2x| - 1\)
Mapping Formula: \((x, y) \to ( \quad\quad\quad , \quad\quad\quad )\)
New Point \((\quad\quad , \quad\quad)\)
Curve Shifters • Algebra II Modular Series Page 1 of 2
Algebra II
Curve Shifters: Synthesis & Application
Student: ______________________
Part 4: Function Synthesis (Build the Transformed Equation)
Write an explicit equation for \(g(x)\) matching the described transformations.
8. Parent: \(f(x) = \sqrt{x}\)
Reflected across the \(x\)-axis, vertically stretched by a factor of 5, shifted left 7 units, and shifted down 3 units.
\(g(x) =\)
9. Parent: \(f(x) = |x|\)
Vertically compressed by a factor of \(\frac{1}{3}\), shifted right 4 units, and shifted up 8 units.
\(g(x) =\)
10. Parent: \(f(x) = x^3\)
Horizontally compressed by a factor of \(\frac{1}{2}\), reflected across the \(y\)-axis, and shifted down 6 units.
\(g(x) =\)
11. Parent: \(f(x) = \frac{1}{x}\)
Vertically stretched by a factor of 4, shifted right 2 units, and shifted up 5 units.
\(g(x) =\)
Part 5: Graphing & Key Attribute Analysis
Plot at least 3 distinct transformed key points. State domain and range.
12. \(g(x) = -2|x + 3| + 5\)
Vertex / Anchor:
Domain:
Range:
Key Points Table:
<table class="w-full text-center border border-slate-300 text-[9px]"><tbody><tr class="bg-slate-100 font-bold border-b border-slate-300"><td class="p-0.5 border-r border-slate-300">\(x\)</td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5"></td></tr><tr><td class="p-0.5 font-bold bg-slate-100 border-r border-slate-300">\(y\)</td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5"></td></tr></tbody></table>
13. \(g(x) = 2\sqrt{x - 1} - 4\)
Endpoint / Anchor:
Domain:
Range:
Key Points Table:
<table class="w-full text-center border border-slate-300 text-[9px]"><tbody><tr class="bg-slate-100 font-bold border-b border-slate-300"><td class="p-0.5 border-r border-slate-300">\(x\)</td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5"></td></tr><tr><td class="p-0.5 font-bold bg-slate-100 border-r border-slate-300">\(y\)</td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5 border-r border-slate-300"></td><td class="p-0.5"></td></tr></tbody></table>
Part 6: Forensic Error Analysis • Factor Before You Shift!
Critical Reasoning [5 pts]
The Scenario: A student analyzes the function \(h(x) = \sqrt{3x - 12} + 2\) and writes in their notes: "The parent function is \(f(x) = \sqrt{x}\). It is horizontally compressed by \(\frac{1}{3}\), shifted right 12 units, and shifted up 2 units."
A. Identify the error & explain why:
B. Factor correctly & state true shift:
Curve Shifters • Algebra II Modular Series Page 2 of 2
Curve Shifters Answer Key
Teacher Answer Key Algebra II • Form & Transformations
Curve Shifters: Complete Solutions
Grading Rubric & Walkthrough
Total: 40 Points
Target Standard: CCSS.MATH.HSF.BF.B.3
Topic: Function Shifts, Flips & Dilations
Official Solutions
Part 1: Master Rule Review • \(g(x) = a \cdot f(b(x - h)) + k\)
\((x, y) \to \left(\frac{x}{b} + h,\; a \cdot y + k\right)\)
\(a\) = Vertical Dilation
\(|a| > 1\) stretch; \(|a| < 1\) shrink. Negative reflects over \(x\)-axis.
\(b\) = Horizontal Dilation
Factor is \(\frac{1}{|b|}\). Negative reflects over \(y\)-axis.
\(h\) = Horizontal Shift
\((x - h)\) moves right; \((x + h)\) moves left by \(h\).
\(k\) = Vertical Shift
\(+k\) moves up; \(-k\) moves down by \(k\) units.
Part 2: Parameter Extraction Answers [16 pts • 1 pt per blank]
Verify both magnitude and direction
1. \(g(x) = -3(x - 4)^2 + 7\) [4 pts]
Parent Function: \(f(x) = x^2\) (Quadratic)
Reflection(s): Reflection across the \(x\)-axis
Stretch / Shrink: Vertical stretch by a factor of 3
Translations: Right 4 units, Up 7 units
2. \(g(x) = \frac{1}{2}\sqrt{x + 5} - 6\) [4 pts]
Parent Function: \(f(x) = \sqrt{x}\) (Square Root)
Reflection(s): None
Stretch / Shrink: Vertical compression by \(\frac{1}{2}\)
Translations: Left 5 units, Down 6 units
3. \(g(x) = 4|-(x - 1)| - 8\) [4 pts]
Parent Function: \(f(x) = |x|\) (Absolute Value)
Reflection(s): Reflection across the \(y\)-axis
Stretch / Shrink: Vertical stretch by a factor of 4
Translations: Right 1 unit, Down 8 units
4. \(g(x) = -\left(\frac{1}{3}x\right)^3 + 2\) [4 pts]
Parent Function: \(f(x) = x^3\) (Cubic)
Reflection(s): Reflection across the \(x\)-axis
Stretch / Shrink: Horizontal stretch by factor of 3
Translations: Up 2 units (no horizontal shift)
Part 3: Coordinate Mapping Solutions [6 pts • 2 pts each]
1 pt for correct rule, 1 pt for mapped point
- For \(f(x) = x^2\), parent point is \((3, 9)\).
\(g(x) = 2(x - 5)^2 - 3\)
Rule: \((x, y) \to (x + 5,\; 2y - 3)\)
\((3 + 5,\; 2(9) - 3) = (8,\; 18 - 3)\)
Transformed \((8, 15)\)
- For \(f(x) = \sqrt{x}\), parent point is \((4, 2)\).
<table class="w-full text-center border border-emerald-300 text-[9px] bg-emerald-50/50"><tbody><tr class="border-b border-emerald-200 font-bold text-slate-800"><td class="p-0.5 border-r border-emerald-200">\(x\)</td><td class="p-0.5 border-r border-emerald-200 font-mono">-4</td><td class="p-0.5 border-r border-emerald-200 font-mono text-rose-700">-3</td><td class="p-0.5 font-mono">-2</td></tr><tr class="font-bold text-slate-800"><td class="p-0.5 border-r border-emerald-200">\(y\)</td><td class="p-0.5 border-r border-emerald-200 font-mono">3</td><td class="p-0.5 border-r border-emerald-200 font-mono text-rose-700">5</td><td class="p-0.5 font-mono">3</td></tr></tbody></table>
<table class="w-full text-center border border-emerald-300 text-[9px] bg-emerald-50/50"><tbody><tr class="border-b border-emerald-200 font-bold text-slate-800"><td class="p-0.5 border-r border-emerald-200">\(x\)</td><td class="p-0.5 border-r border-emerald-200 font-mono text-rose-700">1</td><td class="p-0.5 border-r border-emerald-200 font-mono">2</td><td class="p-0.5 font-mono">5</td></tr><tr class="font-bold text-slate-800"><td class="p-0.5 border-r border-emerald-200">\(y\)</td><td class="p-0.5 border-r border-emerald-200 font-mono text-rose-700">-4</td><td class="p-0.5 border-r border-emerald-200 font-mono">-2</td><td class="p-0.5 font-mono">0</td></tr></tbody></table>
