Graphical intersections,contacts and excluded inputs
Draw both original graphs and interpret all real intersections,including repeated contacts,no-solution cases and contextual exclusions.
Exam scope and assessment notes
Original selected sparse numerical graph teaching with three lessons,twelve lesson/stage groups and sixteen cases. Every curve retains all six source-family coefficients,with at least three zero coefficients. Exact original-input tables and real plotted points and curve strokes precede independent interpretations. Real pointer and keyboard coordinates remain unsnapped;explicit bounded drawing-position tolerances do not soften exact table arithmetic or original zero exclusions. Both original curves remain overlaid for intersection readings;no solutions,repeated contacts,irrational readings,context filtering and cancelled reciprocal exclusions are distinguished. Complete wrong attempts may study a separate explained reference without passing fixed comparisons. Finite table,vertex and segment samples do not prove a globally correct smooth curve. Written reasoning,drawing judgement and model validity need human review. All14Functions source rows have reviewed structural teaching/practice links;other Mathematics B source gaps remain explicit in the current course map;the4Jlink requires these three original graph strands plus actual affine-line construction;4Ktangent construction has a separate original lesson. No official marks,mastery,specialist approval or formal assessment readiness. Single Higher-tier linear4MB1,entry context2026/27only. ISBN9780435044107is metadata only;no verified book body,chapter/Hub mapping or MathematicsA/Foundation/modular inheritance.
Before you begin
- Signed fractions and powers
- Original function domains
- Independent Cartesian scales
- Straight-line graphs
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should be done first when evaluating a signed input in a sparse numerical graph expression?
What you will learn
- Substitute whole signed inputs into every original numerical term.
- Construct exact tables and actual plotted points using the two stated axis scales.
- Retain excluded inputs and separate curve branches.
- Interpret both actual graphs and all original-domain intersections.
Graphical intersections,contacts and excluded inputs: Worked
Start with your own table and actual drawing. The original graph or final interpretation is not supplied. Record each current construction,then its separate interpretation,before studying the explanation. The fixed comparisons do not award marks or mastery.
Draw and interpret both original graphs
Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Read both intersections,including their negative and positive horizontal coordinates. The graphical readings are approximate; the explained reference keeps the exact values separate.
Curve A: y = x^2 − 1
Curve B: y = 1
Horizontal: -3 to 3,step 1/2 dimensionless. Vertical: -12 to 12,step 3 dimensionless.
Drawing-position tolerances for this self-check: horizontal ±0.0375 and vertical ±0.3 coordinate units. Table values and original domain exclusions remain exact. Graphical x-readings use ±0.12 here.
A: y = x^2 − 1 · 0 plotted points · 0 strokesB: y = 1 · 0 plotted points · 0 strokes
My actual table and drawing work
Curve A: y = x^2 − 1
| Given x | My exact output,or undefined |
|---|---|
| -3 | |
| -2 | |
| -1 | |
| -1/2 | |
| 0 | |
| 1/2 | |
| 1 | |
| 2 | |
| 3 |
Enter a curve stroke from keyboard coordinates
Enter your actual successive x,y pairs,one per line. The tool joins only your entered vertices as an approximate sketch. Add detail near bends and keep separate strokes across an excluded input.
Inspect my retained coordinates for A
Plotted points: None yet
Curve B: y = 1
| Given x | My exact output,or undefined |
|---|---|
| -3 | |
| -2 | |
| -1 | |
| -1/2 | |
| 0 | |
| 1/2 | |
| 1 | |
| 2 | |
| 3 |
Enter a curve stroke from keyboard coordinates
Enter your actual successive x,y pairs,one per line. The tool joins only your entered vertices as an approximate sketch. Add detail near bends and keep separate strokes across an excluded input.
Inspect my retained coordinates for B
Plotted points: None yet
Construction status: Not recorded or changed;record again
My separate graph interpretation
Enter your interpretation independently. For intersections,choose none explicitly when appropriate;an empty unsubmitted field does not make that decision. Graphical readings remain approximate. Keep mathematical and context-admitted readings separate.
Interpretation status: Not recorded or changed;record again
Draw and interpret both original graphs
Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Retain both branches of the reciprocal and the original x ≠ 0 condition when reading the line intersections.
Curve A: y = 2/x
Curve B: y = x − 1
Horizontal: -3 to 3,step 1/2 dimensionless. Vertical: -12 to 12,step 3 dimensionless.
Drawing-position tolerances for this self-check: horizontal ±0.0375 and vertical ±0.3 coordinate units. Table values and original domain exclusions remain exact. Graphical x-readings use ±0.12 here.
A: y = 2/x · 0 plotted points · 0 strokesB: y = x − 1 · 0 plotted points · 0 strokes
My actual table and drawing work
Curve A: y = 2/x
| Given x | My exact output,or undefined |
|---|---|
| -3 | |
| -2 | |
| -1 | |
| -1/2 | |
| 0 | |
| 1/2 | |
| 1 | |
| 2 | |
| 3 |
Enter a curve stroke from keyboard coordinates
Enter your actual successive x,y pairs,one per line. The tool joins only your entered vertices as an approximate sketch. Add detail near bends and keep separate strokes across an excluded input.
Inspect my retained coordinates for A
Plotted points: None yet
Curve B: y = x − 1
| Given x | My exact output,or undefined |
|---|---|
| -3 | |
| -2 | |
| -1 | |
| -1/2 | |
| 0 | |
| 1/2 | |
| 1 | |
| 2 | |
| 3 |
Enter a curve stroke from keyboard coordinates
Enter your actual successive x,y pairs,one per line. The tool joins only your entered vertices as an approximate sketch. Add detail near bends and keep separate strokes across an excluded input.
Inspect my retained coordinates for B
Plotted points: None yet
Construction status: Not recorded or changed;record again
My separate graph interpretation
Enter your interpretation independently. For intersections,choose none explicitly when appropriate;an empty unsubmitted field does not make that decision. Graphical readings remain approximate. Keep mathematical and context-admitted readings separate.
Interpretation status: Not recorded or changed;record again
Record every current table and actual drawing.
Written reasoning,curve smoothness,global graph properties and practical model validity need human review. These finite checks do not establish formal assessment readiness or specialist approval. Eleven Functions source rows remain unlinked;4K,4L and4M have separate structural teaching/practice links;this batch does not earn whole4Jcompletion credit.
A common point solves both equations
At an intersection,the two graphs have the same output at the same input. Draw both independently on compatible numbered axes and read every horizontal coordinate at a common point.
Read crossings and contacts
Two graphs can cross,separate entirely,or touch without crossing. A repeated algebraic root can correspond to one geometric point. A sign-change-only search would miss such a contact.
Keep original exclusions after rewriting
Clearing denominators can create candidates at an input that neither original expression permits. Check the original domains before retaining a candidate. Cancellation does not restore an excluded zero.
Explain precision and context
Graphical readings are approximate. Report all mathematical intersections separately from those admitted by a supplied context. An exact reference belongs in the explanation and must not replace the student’s actual drawing or reading.
Sources for this lesson
Pause and explain
Can cancelling a common reciprocal term add zero to the domains of the two original equations?
Worked example
Task 1: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Read both intersections,including their negative and positive horizontal coordinates. The graphical readings are approximate; the explained reference keeps the exact values separate. Original graph(s): y = x^2 − 1; y = 1. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3. Task 2: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Retain both branches of the reciprocal and the original x ≠ 0 condition when reading the line intersections. Original graph(s): y = 2/x; y = x − 1. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3.
Show the worked solution
- Task 1: For y = x^2 − 1, substitute each complete signed input into every original term. The exact reference table is x=-3: 8; x=-2: 3; x=-1: 0; x=-1/2: -3/4; x=0: -1; x=1/2: -3/4; x=1: 0; x=2: 3; x=3: 8.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: For y = 1, substitute each complete signed input into every original term. The exact reference table is x=-3: 1; x=-2: 1; x=-1: 1; x=-1/2: 1; x=0: 1; x=1/2: 1; x=1: 1; x=2: 1; x=3: 1.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 1: Exact mathematical solution reference: -sqrt(2), sqrt(2). Context-admitted graphical readings: -1.414, 1.414.
- Task 1: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
- Task 2: For y = 2/x, substitute each complete signed input into every original term. The exact reference table is x=-3: -2/3; x=-2: -1; x=-1: -2; x=-1/2: -4; x=0: undefined; x=1/2: 4; x=1: 2; x=2: 1; x=3: 2/3.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: The original reciprocal term excludes x=0. Keep negative and positive branch strokes separate; no line through that excluded input is part of this graph.
- Task 2: For y = x − 1, substitute each complete signed input into every original term. The exact reference table is x=-3: -4; x=-2: -3; x=-1: -2; x=-1/2: -3/2; x=0: -1; x=1/2: -1/2; x=1: 0; x=2: 1; x=3: 2.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 2: Exact mathematical solution reference: -1, 2. Context-admitted graphical readings: -1, 2.
- Task 2: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
Answer All mathematical graphical readings: -1.414, 1.414;context-admitted readings: -1.414, 1.414. Exact values: -sqrt(2), sqrt(2). All mathematical graphical readings: -1, 2;context-admitted readings: -1, 2. Exact values: -1, 2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Task 1: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. A cubic may cross a horizontal line three times. Read all intersections rather than the first visible one. Original graph(s): y = x^3 − x; y = 0. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3. Task 2: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Make an explicit no-intersections decision if the original mathematical graphs have no real common point. An empty unsubmitted field is not such a decision. Original graph(s): y = x^2 + 1; y = 0. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3.
Give me a hint
Substitute into every original term,retain exact table values and actual plotted points,keep separate branches at excluded inputs,and enter your interpretation independently.
Compare my reasoning
- Task 1: For y = x^3 − x, substitute each complete signed input into every original term. The exact reference table is x=-3: -24; x=-2: -6; x=-1: 0; x=-1/2: 3/8; x=0: 0; x=1/2: -3/8; x=1: 0; x=2: 6; x=3: 24.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: For y = 0, substitute each complete signed input into every original term. The exact reference table is x=-3: 0; x=-2: 0; x=-1: 0; x=-1/2: 0; x=0: 0; x=1/2: 0; x=1: 0; x=2: 0; x=3: 0.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 1: Exact mathematical solution reference: -1, 0, 1. Context-admitted graphical readings: -1, 0, 1.
- Task 1: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
- Task 2: For y = x^2 + 1, substitute each complete signed input into every original term. The exact reference table is x=-3: 10; x=-2: 5; x=-1: 2; x=-1/2: 5/4; x=0: 1; x=1/2: 5/4; x=1: 2; x=2: 5; x=3: 10.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: For y = 0, substitute each complete signed input into every original term. The exact reference table is x=-3: 0; x=-2: 0; x=-1: 0; x=-1/2: 0; x=0: 0; x=1/2: 0; x=1: 0; x=2: 0; x=3: 0.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 2: Exact mathematical solution reference: no real admitted solutions. Context-admitted graphical readings: none.
- Task 2: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
All mathematical graphical readings: -1, 0, 1;context-admitted readings: -1, 0, 1. Exact values: -1, 0, 1. All mathematical graphical readings: none;context-admitted readings: none. Exact values: none.
Look for these in your work
- My exact table retains every original term and excluded input.
- My plotted points and branch sketches are my own entries on the original independent scales.
- My interpretation is entered separately from my construction.
- My explanation separates mathematical domain,finite display window and supplied context.
- Finite comparison agreement does not establish smoothness,human reasoning or formal assessment readiness.
Task 1: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. The reciprocal-square graph can meet one horizontal line on both sides of its excluded input. Original graph(s): y = 1/x^2; y = 1. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3. Task 2: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. A common point can be a touch without a sign-changing crossing. Read a repeated contact once as a geometric intersection; do not demand a change of sign. Original graph(s): y = x^2; y = 0. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3.
Give me a hint
Substitute into every original term,retain exact table values and actual plotted points,keep separate branches at excluded inputs,and enter your interpretation independently.
Compare my reasoning
- Task 1: For y = 1/x^2, substitute each complete signed input into every original term. The exact reference table is x=-3: 1/9; x=-2: 1/4; x=-1: 1; x=-1/2: 4; x=0: undefined; x=1/2: 4; x=1: 1; x=2: 1/4; x=3: 1/9.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: The original reciprocal term excludes x=0. Keep negative and positive branch strokes separate; no line through that excluded input is part of this graph.
- Task 1: For y = 1, substitute each complete signed input into every original term. The exact reference table is x=-3: 1; x=-2: 1; x=-1: 1; x=-1/2: 1; x=0: 1; x=1/2: 1; x=1: 1; x=2: 1; x=3: 1.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 1: Exact mathematical solution reference: -1, 1. Context-admitted graphical readings: -1, 1.
- Task 1: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
- Task 2: For y = x^2, substitute each complete signed input into every original term. The exact reference table is x=-3: 9; x=-2: 4; x=-1: 1; x=-1/2: 1/4; x=0: 0; x=1/2: 1/4; x=1: 1; x=2: 4; x=3: 9.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: For y = 0, substitute each complete signed input into every original term. The exact reference table is x=-3: 0; x=-2: 0; x=-1: 0; x=-1/2: 0; x=0: 0; x=1/2: 0; x=1: 0; x=2: 0; x=3: 0.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 2: Exact mathematical solution reference: 0. Context-admitted graphical readings: 0.
- Task 2: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
All mathematical graphical readings: -1, 1;context-admitted readings: -1, 1. Exact values: -1, 1. All mathematical graphical readings: 0;context-admitted readings: 0. Exact values: 0.
Look for these in your work
- My exact table retains every original term and excluded input.
- My plotted points and branch sketches are my own entries on the original independent scales.
- My interpretation is entered separately from my construction.
- My explanation separates mathematical domain,finite display window and supplied context.
- Finite comparison agreement does not establish smoothness,human reasoning or formal assessment readiness.
Task 1: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Two hypothetical dimensionless model quantities are admitted only for input 1/2 to 2. First record all mathematical intersections,then the context-admitted list separately. No physical law is established. Original graph(s): y = 1/x^2; y = 1. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3. Task 2: Draw both original mathematical graphs separately from your own value tables and plotted points. Retain every branch and read all real intersections,then distinguish those admitted by the stated context. Both supplied hypothetical expressions exclude zero. Cancelling common reciprocal terms cannot add zero to either original domain. State a no-solution decision if the only cleared-equation candidate was excluded. Original graph(s): y = x + 1/x; y = 1/x. Table inputs: -3, -2, -1, -1/2, 0, 1/2, 1, 2, 3.
Give me a hint
Substitute into every original term,retain exact table values and actual plotted points,keep separate branches at excluded inputs,and enter your interpretation independently.
Compare my reasoning
- Task 1: For y = 1/x^2, substitute each complete signed input into every original term. The exact reference table is x=-3: 1/9; x=-2: 1/4; x=-1: 1; x=-1/2: 4; x=0: undefined; x=1/2: 4; x=1: 1; x=2: 1/4; x=3: 1/9.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: The original reciprocal term excludes x=0. Keep negative and positive branch strokes separate; no line through that excluded input is part of this graph.
- Task 1: For y = 1, substitute each complete signed input into every original term. The exact reference table is x=-3: 1; x=-2: 1; x=-1: 1; x=-1/2: 1; x=0: 1; x=1/2: 1; x=1: 1; x=2: 1; x=3: 1.
- Task 1: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 1: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 1: Exact mathematical solution reference: -1, 1. Context-admitted graphical readings: 1.
- Task 1: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
- Task 2: For y = x + 1/x, substitute each complete signed input into every original term. The exact reference table is x=-3: -10/3; x=-2: -5/2; x=-1: -2; x=-1/2: -5/2; x=0: undefined; x=1/2: 5/2; x=1: 2; x=2: 5/2; x=3: 10/3.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: The original reciprocal term excludes x=0. Keep negative and positive branch strokes separate; no line through that excluded input is part of this graph.
- Task 2: For y = 1/x, substitute each complete signed input into every original term. The exact reference table is x=-3: -1/3; x=-2: -1/2; x=-1: -1; x=-1/2: -2; x=0: undefined; x=1/2: 2; x=1: 1; x=2: 1/2; x=3: 1/3.
- Task 2: Plot these original points using horizontal step 1/2 and vertical step 3. Retain finite off-window coordinates instead of moving them onto the boundary. Use extra points near bends; a finite table or sketch does not prove an entire smooth curve.
- Task 2: The original reciprocal term excludes x=0. Keep negative and positive branch strokes separate; no line through that excluded input is part of this graph.
- Task 2: Read the horizontal coordinate at every common point of the two graphs,including a touch that does not cross. Compare these approximate readings with the original equations and keep the exact reference separate.
- Task 2: Exact mathematical solution reference: no real admitted solutions. Context-admitted graphical readings: none.
- Task 2: The private reference factor certificate multiplies to the complete original cleared equation; original zero exclusions are restored afterward. It is a check for this authored example,not a claim that a sparse graph table proves all roots of an arbitrary equation.
All mathematical graphical readings: -1, 1;context-admitted readings: 1. Exact values: -1, 1. All mathematical graphical readings: none;context-admitted readings: none. Exact values: none.
Look for these in your work
- My exact table retains every original term and excluded input.
- My plotted points and branch sketches are my own entries on the original independent scales.
- My interpretation is entered separately from my construction.
- My explanation separates mathematical domain,finite display window and supplied context.
- Finite comparison agreement does not establish smoothness,human reasoning or formal assessment readiness.
Common mistakes
- Replacing the original expression with only one of its terms.
- Reading independent axis scales as equal picture lengths.
- Joining branches across an input excluded by an original reciprocal term.
- Clamping an off-window point onto the drawing edge.
- Treating an empty unsubmitted answer as an explicit no-solution decision.
- Assuming a finite self-check proves a globally smooth curve or validates a physical model.
What can you explain now?
Can cancelling a common reciprocal term add zero to the domains of the two original equations?
Compare with the explanation
No. Both original domains must still be respected;an excluded candidate is not a solution.
Use the original expression and original context conditions before interpreting the drawing.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Close the reference. Rebuild your original-input table and actual graph,then justify domain,window and context decisions. Read every intersection from both retained drawings and restore all original exclusions.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.