Higher: use cubic intersections and difference graphs
Draw genuine cubic and line pairs, connect all common inputs to difference zeros, and distinguish crossing, repeated contact and clipped intersections.
Exam scope and assessment notes
Additional Higher only. Higher includes shared Foundation content. The Graphs pathway has nine shared lessons, two Higher line-equation lessons, four graph-family lessons, nine transformation lessons and these three graphical-method lessons. H:3.3D requires drawing a tangent and using a large signed rise/run triangle with actual scales and units; differentiating is not a substitute and section3.4 calculus is not credited. H:3.3E requires both line/quadratic and genuine line/cubic intersection lessons, all practice stages, paired original coordinates and the zeros of nonlinear-minus-linear. A zero on the difference graph does not replace an original intersection height. A repeated zero represents one distinct point. The nine Foundation and seven additional Higher Graphs source rows now have teaching and staged-practice links; these are links, not exhaustive subskill approval, mastery or examination readiness. The wider Maths A course still has unlinked requirements. Graph constructions and readings are approximate where stated; exact fixed-model calculations are separate checks. Written work is not automatically marked. Acknowledging a paper attempt or revealing a reference does not prove correct construction or award marks. No general solver, official mark scheme or validated cumulative assessment is supplied. Specialist review and formal assessment validation remain unestablished. Pearson’s chapter-level modular map and uncorrected-proof navigation stay separate evidence: Graphs9 map starts386 while the proof chapter starts385, and neighbouring calculus topics do not replace H3.3D. Final full-book/Teaching Hub bodies and pagination alignment remain unverified.
Before you begin
- Generate and draw genuine cubic and straight-line graphs.
- Subtract complete expressions and use simple factorisation as a post-graph check.
- Read signed and fractional coordinate pairs and interpret finite plotting windows.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
The difference factors as d=x(x−2)(x+3). Which list retains every distinct zero input?
What you will learn
- Find common points of horizontal or oblique lines with genuine cubic graphs.
- Use y₂−y₁=0 while retaining negative, zero and fractional inputs and paired original heights.
- Classify crossing and repeated touching roots, and interpret a unique approximate intersection within a finite model domain.
y₁=2 and y₂=x³−3x+2. Shared original domain: all real inputs. Blue is the linear function; red is the nonlinear function. Keep both original coordinates at each common point.
Draw the original pair and subtract the linear function from the nonlinear function on paper. No input or solved intersection is selected for you. The two panels share the same horizontal scale; their vertical scales are independent.
Horizontal view: −2.3 to 2.3. Original-pair vertical view: −4 to 8; difference vertical view: −6 to 6. A finite crop neither proves nonintersection nor changes a valid coordinate or domain. Points outside a panel are omitted rather than moved to a border.
No input is preselected. Choose your own reading from the graph.
Plot a genuine cubic with both turns. Keep the zero input; cancelling x would lose one common point.
A genuine cubic can meet a line several times
A cubic graph has a nonzero x³ coefficient. Plot the complete stated expression, not a quadratic substitute or only the values at suspected intersections. Use enough points to show the changes in direction relevant to the view, including points between crossings. Draw the line and smooth cubic on one labelled panel, and read each common input with its own common height.
A horizontal line is still a linear graph: when it is y₁=2, every original common height is 2. An oblique line gives different heights at different inputs. Common inputs can be negative, zero or fractional. Do not cancel a factor x when analysing the difference, because doing so can remove the genuine input x=0.
Link each crossing or contact to a difference zero
Form d(x)=y₂(x)−y₁(x) by subtracting the whole line. Draw this genuine cubic on an aligned panel using the same horizontal scale and a clear d=0 baseline. At every common input r, the original pair has one shared height y₁(r), while the difference point is (r,0). Link all such inputs between panels without replacing the original heights.
Use signs on either side of a zero to distinguish the local picture. A simple zero in these examples changes the difference sign and gives a crossing. A repeated squared factor can give a touching contact without a sign change, but it is still only one distinct coordinate pair. A factorisation is a useful check after drawing; the lesson does not ask for a general algorithm to solve every cubic.
Use the domain, scale and accuracy honestly
A finite window can hide a common height even when its input is inside the horizontal view. Widen the height bounds and preserve the coordinates. A separately scaled difference panel may reveal the root input, but that does not excuse omitting the original pair: the paired common reading still belongs to the original graphs. Restrict both graphs and the difference to the common model domain when one is stated.
Give graph-derived irrational inputs approximately at the requested precision. Exact rational pairs may be retained when the grid and original equations support them. A rounded input does not have to reproduce the common height exactly when substituted back: the rounding changed it. For a strictly increasing cube rule on a nonnegative time interval, a horizontal target can have only one common input; a cropped or empty view by itself proves no such uniqueness claim.
Sources for this lesson
Pause and explain
Two original graphs give d=(x−2)²(x+1). What occurs at x=2 in these graphs?
Worked example
Plot y₁=2 and y₂=x³−3x+2 on −2.3≤x≤2.3 with −4≤y≤8. Plot d=y₂−y₁ in an aligned panel over the same x-window with −6≤d≤6 and a clear zero baseline. Draw the actual smooth cubic with both visible bends, locate all three common points, and give their graphical x-estimates to one decimal place. Link each to the corresponding zero point on d and keep its original paired height.
Show the worked solution
- Draw the horizontal line y₁=2. For the cubic, the heights at x=−2,−1,0,1,2 are 0,4,2,0,4. Additional half-unit points (−3/2,25/8), (−1/2,27/8), (1/2,5/8), (3/2,7/8) help resolve the smooth bends and separate crossings. Plot the complete cubic expression rather than connecting only the suspected common points.
- Subtract the line: d=(x³−3x+2)−2=x³−3x. Its heights at x=−2,−1,0,1,2 are −2,2,0,−2,2. Draw the smooth cubic in the difference panel and locate three zeros: about −1.7, exactly 0 from the equations, and about 1.7. A post-attempt check d=x(x²−3) confirms the negative, zero and positive zero inputs.
- The original common height is 2 in all three cases. Report approximately (−1.7,2), the exact pair (0,2), and approximately (1.7,2). Their difference partners are approximately (−1.7,0), exactly (0,0), and approximately (1.7,0). The nonzero inputs are graph estimates to one decimal place; their rounded values are not exact roots.
Answer Original common points: approximately (−1.7,2), exactly (0,2), approximately (1.7,2). Difference d=x³−3x=x(x²−3) has the corresponding three zero inputs, with difference height zero.
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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.
Plot y₁=x+1 and y₂=2x³+3x²−x+1 on −2.5≤x≤1 with −12≤y≤7. Form d=y₂−y₁ before revealing its trace, and draw it in an aligned panel using the same x-window, −10≤d≤5 and a zero baseline. Read every common point and check its original paired ordinate. Use half-unit x ticks, include a point between x=0 and x=1/2, and keep exact fractions where the grid/data support them.
Give me a hint
Subtract all of x+1. Keep any factor x instead of dividing it away, and use the original line to recover the height for each common input.
Compare my reasoning
- Draw the original line and genuine cubic. Useful cubic points are (−5/2,−9), (−2,−1), (−3/2,5/2), (−1,3), (−1/2,2), (0,1), (1/4,31/32), (1/2,3/2), (1,5). The point at x=1/4 helps show the cubic below the line between two nearby common inputs; do not replace that portion by a straight join between them.
- The difference is (2x³+3x²−x+1)−(x+1)=2x³+3x²−2x. Plot it on the aligned panel. After attempting the graph, factor d=x(2x−1)(x+2), giving the three distinct inputs −2,0,1/2. Retaining the factor x is essential.
- The original line gives heights −1,1,3/2 respectively. Mark original common points (−2,−1), (0,1), (1/2,3/2), linked to difference points (−2,0), (0,0), (1/2,0). Check each input in the cubic too; all three contacts cross because each corresponding simple zero changes the difference sign.
Common points: (−2,−1), (0,1), (1/2,3/2). Difference d=2x³+3x²−2x=x(2x−1)(x+2), with zero points (−2,0), (0,0), (1/2,0).
Look for these in your work
- Draw a genuine cubic with the between-root sample, the line and the aligned difference trace.
- Retain negative, zero and fractional common inputs.
- Keep all original coordinate pairs correctly paired and link each to its difference zero.
Compare y₁=−x+2 and y₂=2x³−3x²−x+3. Draw the original pair on −1≤x≤2 with −2≤y≤6, then form and draw their difference in a separate aligned panel on the same horizontal window with −5≤d≤6 and a zero baseline. Use half-unit x and y ticks. Find all distinct common coordinate pairs, classify each as crossing or touching, and explain the count. Attempt both panels before opening any solved factors or coordinates.
Give me a hint
Keep the subtraction bracketed. Look at the difference sign on each side of a zero, and use the original line to recover each common height. A repeated zero is still one input.
Compare my reasoning
- Draw the original pair and your difference graph before checking the algebra. The difference is d=2x³−3x²+1. At x=−1,−1/2,0,1/2,1,3/2,2 it has heights −4,0,1,1/2,0,1,5. The original cubic heights at those inputs are −1,5/2,3,2,1,3/2,5. These samples show a crossing on the left and a later contact without replacing the cubic by a quadratic.
- After the attempt, d=(x−1)²(2x+1). Thus the distinct common inputs are −1/2 and 1. At x=−1/2 the factor 2x+1 changes sign while the squared factor is positive, so this is a crossing. Near x=1 the remaining factor is positive and the square stays nonnegative, so the difference touches zero without changing sign.
- Substitute into y₁=−x+2: the original common points are (−1/2,5/2) and (1,1). Link them to (−1/2,0) and (1,0) in the difference panel. The repeated input x=1 is one touching point, not two separate intersections. The factorisation accounts for every real common input in these full-domain functions.
Two distinct common points: (−1/2,5/2), crossing; (1,1), touching. Difference d=2x³−3x²+1=(x−1)²(2x+1), with distinct zero points (−1/2,0), (1,0).
Look for these in your work
- Attempt the original pair and difference before viewing the solved factors.
- Give both correctly paired original points, including the negative fractional input.
- Use the difference sign and multiplicity to distinguish one crossing from one repeated touching contact.
Two invented sensor models use A(t)=20 and B(t)=t³+1 for 0≤t≤4 seconds, both measured in sensor units (su). An initial reading window 0≤y≤6 shows no common reading. Draw the original pair with t (s) horizontally and reading (su) vertically, widen to 0≤y≤70, and keep the full permitted time interval. Draw d(t)=B(t)−A(t) in an aligned panel over 0≤t≤4 with −20≤d≤50 and a zero baseline. Estimate the time of agreement to one decimal place, give its paired common reading, and justify whether it is unique.
Give me a hint
The horizontal target reading can be hidden by the first height crop even when its time lies in view. Subtract A from B and use the increase of t³ on the stated nonnegative interval to assess uniqueness.
Compare my reasoning
- The initial crop hides the horizontal A=20 and therefore cannot show the original common point. In the expanded view, draw A=20 and the actual cubic B on 0≤t≤4. B has readings 1,2,9,28,65 at t=0,1,2,3,4. Its reading crosses 20 between 2 and 3 seconds; do not alter the model domain when widening the height scale.
- The difference is d=t³−19. Draw it on the aligned panel. At t=2.6 its value is −1.424 and at t=2.7 it is 0.683, bracketing the zero; the original B readings are 18.576 and 20.683. A one-decimal graph estimate is t≈2.7 s, with common original reading 20 su and difference height 0. The rounded time 2.7 is not an exact root and need not reproduce exactly 20 when substituted.
- The cube rule is strictly increasing on 0≤t≤4, so it can meet the constant 20 only once; the values at 2 and 3 seconds establish that it does meet within the domain. Label the approximate original pair (2.7 s,20 su) and corresponding difference point (2.7 s,0 su). Use the scale honestly: a more precise numerical model check would not make the freehand graph accurate to extra decimal places.
The models agree once, at approximately t=2.7 s with reading 20 su. The corresponding difference zero is approximately (2.7 s,0 su) for d=t³−19. The initial height crop hid the meeting; strict increase on the model domain supports uniqueness.
Look for these in your work
- Draw both original models and the aligned difference over the stated time domain.
- Report the approximate time with its correctly paired common reading and units.
- Explain the cropped view, rounded input and uniqueness without claiming exact graph accuracy.
Common mistakes
- Replacing a genuine cubic by a quadratic sketch, omitting its bends, or cancelling x and losing a zero input.
- Mixing the x-value of one crossing with the original height of another, or using zero as every original common height.
- Counting a repeated root twice, treating a crop as nonexistence, or presenting a rounded graph input as exact.
What can you explain now?
If a line and cubic share the original point (1/2,−4), what is the corresponding point on d=y₂−y₁, and does it mean the original height was zero?
Compare with the explanation
The difference point is (1/2,0); the original common height remains −4.
At the same input, (−4)−(−4)=0. The difference ordinate measures the separation, not either original height.
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Make it stick
Explain how three cubic common points, a repeated contact and a unique approximate crossing appear on the original pair and on its difference graph. State why zero and fractional roots and the original paired heights must be retained.
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.