Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
| | Graph 4
Piecewise Steps
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
| | Graph 5
Sideways Parabola
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
|
Box It Up! Domain & Range Note Catcher • Page 1 of 2
Complete analysis for Graphs 6–10, analyze common misconceptions, and design a custom graph.
Recording Sheet • Part 2
Name: ___________________________
Check your endpoints: [ ] includes, ( ) excludes!
| Graph & Type | Bounding Box Walls | Domain (\(x\)) | Range (\(y\)) | Function? & Justification |
|---|---|---|---|---|
| Graph 6 | ||||
| Linear Ray |
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
| | Graph 7
Cubic S-Curve
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
| | Graph 8
Vertical Segment
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
| | Graph 9
V-Shape Peak
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
| | Graph 10
Overlapping Steps
|
Left: ________
Right: ________
Bottom: ________
Top: ________
|
Inequality:
Interval:
|
Inequality:
Interval:
|
[ ] Yes [ ] No
Why? (VLT reasoning):
|
Error Analysis Spotlight
A classmate evaluated Graph 6 (Linear Ray) and claimed:
“The domain is \([-4, \infty)\) because the line starts at \(-4\), and the range is \((-\infty, \infty)\) because it has an arrow going down.”
Find and correct the TWO mistakes the classmate made:
Challenge: Sketch Your Relation
Sketch a graph with Domain: \([-4, 5)\), Range: \([-2, 4]\), and that is NOT a function.
x y -4 5 4 -2
Proof it fails VLT: At \(x = \) ____, there are points at \(y = \) ____ and \(y = \) ____.
Box It Up! Domain & Range Note Catcher • Page 2 of 2
Bounded circle height
| NO, Not a Function
Fails VLT; e.g., the vertical line \(x = 0\) intersects circle at both \((0, 4)\) and \((0, -4)\).
| | Graph 4
Piecewise Steps
|
Left: \(-6\) (solid)
Right: \(5\) (solid)
Bottom: \(-3\) (open)
Top: \(5\) (solid)
|
Ineq: \(-6 \le x \le 5\)
Int: \([-6, 5]\)
At \(x = -1\), solid dot includes \(-1\)
|
Ineq: \(-3 < y \le 5\)
Int: \((-3, 5]\)
Union of \((-3, 2]\) and \([1, 5]\)
| YES, Function
Passes VLT; at \(x = -1\), the line hits only solid point \((-1, 1)\) because \((-1, -3)\) is open.
| | Graph 5
Sideways Parabola
|
Left: \(-3\) (vertex)
Right: \(\infty\)
Bottom: \(-\infty\)
Top: \(\infty\)
|
Ineq: \(x \ge -3\)
Int: \([-3, \infty)\)
Starts at vertex \(x = -3\)
|
Ineq: \(-\infty < y < \infty\)
(All Real Numbers \(\mathbb{R}\))
Int: \((-\infty, \infty)\)
| NO, Not a Function
Fails VLT; e.g., line \(x = 1\) intersects at \((1, 3)\) and \((1, -1)\).
|
Common Student Misconceptions (Graphs 1–5):
• Graph 1: Students often assume arrows pointing up don't expand horizontally. Remind them that parabolas continue widening infinitely in both \(x\)-directions.
• Graph 4: Watch for students writing a split domain like \([-6, -1) \cup [-1, 5]\). Since \(x = -1\) is defined by the solid dot, the domain connects seamlessly as \([-6, 5]\).
Box It Up! Domain & Range Teacher Answer Key • Page 1 of 2
Authoritative solutions for Graphs 6–10, Error Analysis diagnosis, and Challenge rubric.
Teacher Resource • Page 2
Key Reminder: Vertical line \(x = c\) has domain \(\{c\}\) and fails VLT. Discrete ranges must use set notation \(\{a, b\}\).
Graphs 6–10
| Graph & Type | Bounding Box Walls | Domain (\(x\)) | Range (\(y\)) | Function? & Justification |
|---|---|---|---|---|
| Graph 6 | ||||
| Linear Ray |
|
Left: \(-4\) (open)
Right: \(\infty\)
Bottom: \(-\infty\)
Top: \(5\) (open)
|
Ineq: \(x > -4\)
Int: \((-4, \infty)\)
Open endpoint \(\circ \implies (\)
|
Ineq: \(y < 5\)
Int: \((-\infty, 5)\)
Open ceiling \(\circ \implies )\)
| YES, Function
Passes VLT; linear ray has exactly one output for each \(x > -4\).
| | Graph 7
Cubic S-Curve
|
Left: \(-4\) (solid)
Right: \(4\) (open)
Bottom: \(-4\) (solid)
Top: \(4\) (open)
|
Ineq: \(-4 \le x < 4\)
Int: \([-4, 4)\)
Mixed: solid left, open right
|
Ineq: \(-4 \le y < 4\)
Int: \([-4, 4)\)
Floor included, ceiling excluded
| YES, Function
Passes VLT; every vertical line hits curve at most once.
| | Graph 8
Vertical Segment
|
Left: \(3\) (solid)
Right: \(3\) (solid)
Bottom: \(-5\) (solid)
Top: \(4\) (solid)
|
Ineq: \(x = 3\)
Set: \(\{3\}\)
Single value only
|
Ineq: \(-5 \le y \le 4\)
Int: \([-5, 4]\)
Continuous interval of \(y\)
| NO, Not a Function
Fails VLT completely; the line \(x = 3\) intersects infinitely many points.
| | Graph 9
V-Shape Peak
|
Left: \(-6\) (open)
Right: \(5\) (solid)
Bottom: \(-3\) (solid)
Top: \(5\) (solid)
|
Ineq: \(-6 < x \le 5\)
Int: \((-6, 5]\)
Continuous across peak
|
Ineq: \(-3 \le y \le 5\)
Int: \([-3, 5]\)
From floor \(-3\) to peak \(5\)
| YES, Function
Passes VLT; each input has one output (absolute value shape).
| | Graph 10
Overlapping Steps
|
Left: \(-5\) (solid)
Right: \(6\) (solid)
Bottom: \(-3\) (solid)
Top: \(3\) (solid)
|
Ineq: \(-5 \le x \le 6\)
Int: \([-5, 6]\)
Segments overlap on \([0, 2]\)
|
Set: \(\{-3, 3\}\)
NOT an interval!
Only two discrete \(y\)-values exist
| NO, Not a Function
Fails VLT; for any \(x \in [0, 2]\), there are two outputs: \(y = 3\) and \(y = -3\).
|
Error Analysis Answer Key
Classmate's claims for Graph 6: Domain \([-4, \infty)\) and Range \((-\infty, \infty)\).
Mistake 1 (Domain Bracket): Point \((-4, 5)\) is an open circle, so \(-4\) is NOT included. The correct domain is \((-4, \infty)\) or \(x > -4\).
Mistake 2 (Range Ceiling): The ray only travels downward; it has an upper boundary at \(y = 5\) (open) and does not go to \(+\infty\). The correct range is \((-\infty, 5)\) or \(y < 5\).
Award full credit if students identify both the open-circle error and the range ceiling.
Challenge Criteria & Rubric
Student graph criteria: Domain: \([-4, 5)\), Range: \([-2, 4]\), NOT a function.
• Domain Check: Solid point at \(x = -4\) and open circle at \(x = 5\); graph spans completely between them.
• Range Check: Lowest point reaches \(y = -2\) and highest reaches \(y = 4\) (both included).
• VLT Check: Must contain at least one vertical line with \(\ge 2\) points (e.g. an oval, vertical segment, or overlapping branches).
Acceptable drawings include sideways parabolas, ovals, piecewise overlaps, or an S-curve that loops back.
Box It Up! Domain & Range Teacher Answer Key • Page 2 of 2
