Quadratic Models (AI SL)
A quadratic model uses \(f(x)=ax^2+bx+c\) to describe a quantity that rises to a peak (or falls to a trough) and then reverses - projectile heights, areas enclosed by a fixed perimeter, revenue against price, and more. This topic covers setting up a quadratic from a real context, reading off its vertex, roots and intercepts, and using it to answer questions about the situation it models.
What the syllabus says
This topic maps onto the Functions & Modelling unit of the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL2.5 | Modelling with quadratic functions \(f(x)=ax^2+bx+c\), \(a\neq0\). Using technology to find roots. Axis of symmetry, vertex, zeros and roots, intercepts on the \(x\)-axis and \(y\)-axis. Contexts include cost functions, satellite dishes, bridges and projectile motion. |
| SL2.6 | Modelling skills: given a context, recognise and choose an appropriate model; determine a reasonable domain; find the parameters of a model by solving equations simultaneously or from given conditions; use technology to fit and test the model. |
Fitting models by regression (SL2.6) is covered alongside statistics, but the modelling process applies directly to quadratics here.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a quadratic model?
A quadratic model is a function \(f(x)=ax^2+bx+c\) (\(a\neq0\)) used to describe a real quantity that changes with a single turning point - it either has a maximum (if \(a<0\)) or a minimum (if \(a>0\)). The parameters \(a\), \(b\) and \(c\) are chosen to fit the context.
e.g. \(A(x)=x(80-2x)\) models the area of a fenced pen of width \(x\); it is quadratic because expanding gives \(-2x^2+80x\).
What is the vertex of a parabola?
The vertex is the turning point of the parabola - its maximum if the curve opens downward, or minimum if it opens upward. Its \(x\)-coordinate is \(x=-\dfrac{b}{2a}\), found either algebraically or by using the GDC's maximum/minimum graph tool.
e.g. For \(A(x)=-2x^2+80x\), the vertex is at \(x=-\dfrac{80}{2(-2)}=20\), giving \(A(20)=800\).
What are the roots (zeros) of a model?
The roots are the values of \(x\) where the function equals zero - where the graph crosses the \(x\)-axis. In context, they often mark the start and end of a physically meaningful interval, such as where a projectile lands or where profit breaks even.
e.g. For \(P(x)=-x^2+50x-400\), solving \(P(x)=0\) gives \(x=10\) and \(x=40\) - the break-even points.
What is a reasonable domain?
A reasonable domain is the set of \(x\)-values over which a model can sensibly be trusted, given the real-world constraints of the situation - for example, a width can't be negative, and a model fitted to data shouldn't be extrapolated far beyond it.
e.g. For a ball-height model built from data at \(t=0,2,5\) seconds, a reasonable domain is \(0\le t\le5\).
What is optimisation with a quadratic model?
Optimisation means using the vertex of a quadratic model to find the best possible outcome in context - the maximum area, the maximum revenue, or the minimum cost - rather than just evaluating the function at a single point.
e.g. For \(R(x)=(20+x)(500-20x)\), the vertex at \(x=2.5\) gives the price \($22.50\) that maximises revenue at \($10\,125\).
Key formulas
A handful of results cover almost every quadratic-modelling question. The tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The general form and vertex formula are standard results you're expected to know; the quadratic formula for finding roots is on the official formula booklet.
| Formula | Used for | Booklet? |
|---|---|---|
| \(f(x)=ax^2+bx+c\) | General quadratic model | Not in booklet |
| \(x=-\dfrac{b}{2a}\) | \(x\)-coordinate of the vertex | Not in booklet |
| \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Quadratic formula, for roots | ✓ Yes |
| \(y\)-intercept \(=c\) | Value of the model at \(x=0\) | Not in booklet |
Vertex form vs general form
The same parabola can be written two ways - each makes different features easy to read off directly.
| Feature | General form \(ax^2+bx+c\) | Vertex form \(a(x-h)^2+k\) |
|---|---|---|
| Easy to read off | \(y\)-intercept (\(c\)) | Vertex \((h,k)\) directly |
| Finding the vertex | Compute \(x=-\dfrac{b}{2a}\), then substitute | Already given as \((h,k)\) |
| Direction of opening | Sign of \(a\) | Sign of \(a\) |
| Typical AI SL use | Fitting a model to raw data | Describing a known shape (e.g. an arch) |
Setting up a model from data
When you're given data points instead of an equation, you build the model by substitution.
Three points, three unknowns
Substitute each \((x,y)\) pair into \(f(x)=ax^2+bx+c\) to get three simultaneous equations in \(a\), \(b\) and \(c\).
Solve with your GDC's equation solver rather than by hand.
Regression from many points
With more than three data points, use quadratic regression on the GDC to find the best-fitting \(a\), \(b\) and \(c\) directly.
Check \(R^2\) to see how well the model fits.
Given conditions
Sometimes the model is built from a stated rule, such as "each $1 rise sells 20 fewer tickets" - translate this into an expression before expanding.
Expand and simplify before finding the vertex.
Reading and using the model
Once you have \(a\), \(b\) and \(c\), most questions ask you to extract a feature or evaluate the model at a point.
Maximum or minimum
Find the vertex to answer "what is the greatest/least value" questions - area, revenue, height, profit.
Not in the formula booklet - use \(x=-b/(2a)\) or the GDCZeros in context
Solve \(f(x)=0\) to answer "when does it start/end/break even" questions - landing time, break-even units, ground level.
Solve with the GDC or the quadratic formulaEvaluating at a point
Substitute a given \(x\) directly to answer "what is the value at..." questions, then compare it to the maximum if asked.
Direct substitutionWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A ball is thrown into the air, and its height \(y\) (in metres) is recorded at time \(x\) (in seconds). The table shows measured values.
| \(x\) | 0 | 2 | 5 |
|---|---|---|---|
| \(y\) | 20 | 44 | 35 |
The relationship is modelled by a quadratic function \(y=ax^2+bx+c.\)
(a) Use the data to set up and solve a system of equations to find the values of \(a,\ b\) and \(c.\)
(b) State a reasonable domain for this model.
(c) Using a GDC, find the coordinates of the vertex of the graph, and interpret its meaning in context.
Worked solution
(a) Substituting the three data points: \(-3(0)^2+18(0)+c=20,\) etc. Solving simultaneously gives \(a=-3,\ b=18,\ c\) M1
\(=20.\) A1
(b) A reasonable domain is \(0 \le x \le 5,\) since values outside this range are not supported by the data and the model is only reliable within it. R1 A1
(c) Using a GDC, the vertex is at \((3, 47).\) M1
This represents the maximum value of \(y\) predicted by the model. R1
A parabolic arch has height \(h(x)=-0.05x^{2}+2x\) m, where \(x\) is the horizontal distance in metres from one base.
(a) Find the height at \(x=10.\)
(b) Find the width of the arch at ground level.
(c) Find the maximum height.
(d)(i) Find the value of \(x<20\) where the height is \(15\) m.
(d)(ii) Find the value of \(x>20\) where the height is \(15\) m.
Worked solution
(a) Height at \(x=10\).
\(h(10)=-0.05(10)^2+2(10)=-5+20=15\text{ m}.\) A1
(b) Width at ground level. Set \(h=0\): \(x(-0.05x+2)=0\Rightarrow x=0\) or \(x=40\). Width \(=40\) m. A1
(c) Maximum height. Vertex midway at \(x=20\):
\(h(20)=-0.05(20)^2+2(20)=-20+40=20\text{ m}.\) A1
(d)(i) \(-0.05x^2+2x=15\Rightarrow x^2-40x+300=0\Rightarrow x=\dfrac{40\pm20}{2}\) M1
\(x=10\) A1
(d)(ii) \(x=30.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Forgetting to state a domain. Modelling questions at AI SL often carry a specific mark for stating a reasonable domain (e.g. \(0\le x\le5\)) - leaving this out loses easy marks even when the maths is correct.
- Confusing the vertex's \(x\)-value with the maximum/minimum value. \(x=-b/(2a)\) gives where the turning point occurs, not the turning point's height - you must substitute back in to get the actual maximum or minimum value.
- Not expanding a bracketed model before finding the vertex. A model like \(R(x)=(20+x)(500-20x)\) must be expanded to \(ax^2+bx+c\) form first, otherwise \(a\) and \(b\) can't be read off directly.
- Extrapolating far beyond the data range. A quadratic fitted to a handful of points can look convincing but becomes unreliable outside the range it was built from - always sense-check predictions against the reasonable domain.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
The heart of AI modelling - find the best-fitting curve for a data set, not just a straight line.
- Enter the data in two lists (x and y).
- STAT → EDIT to enter L1/L2, then STAT → CALC → QuadReg / CubicReg / ExpReg / PwrReg / SinReg.TI-84
- In a Lists & Spreadsheet page enter the data, then menu → Statistics → Stat Calculations → choose the regression type.Nspire
- Main menu → Statistics, enter the data in lists, then CALC → REG and pick X² / X³ / Exp / Power / Sin.Casio
Tip: Turn DiagnosticOn (TI-84: 2nd → 0 → DiagnosticOn) to see \(R^2\). Choose the model with the best \(R^2\) that also makes sense for the context.
After fitting several models (linear, quadratic, exponential...), you need to decide which fits the data best - \(R^2\) is the key tool.
- Fit each candidate model in turn and note the \(R^2\) value each time.
- Turn DiagnosticOn first (2nd → 0, scroll to DiagnosticOn, ENTER) - then \(R^2\) appears after every regression.TI-84
- \(R^2\) is shown automatically after each regression calculation in the Statistics menu.Nspire
- \(R^2\) (displayed as r²) appears in the regression output; run CALC → REG for each model type and compare.Casio
- The model with \(R^2\) closest to 1 explains the most variation in \(y\) - but also consider whether the model makes sense for the context.
Tip: \(R^2\) alone doesn't tell you whether the model is appropriate - always look at the scatter plot too. A high \(R^2\) on a model that shouldn't apply is meaningless.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Quadratic modelling questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What does a quadratic model actually represent?
A quadratic model, \(f(x)=ax^2+bx+c\), describes a quantity that rises then falls (or falls then rises) - like the height of a thrown ball, the area enclosed by a fixed length of fencing, or revenue against price. The parabola's vertex marks the turning point of that quantity.
How do I find the parameters a, b and c from data?
If you're given three data points, substitute each into \(f(x)=ax^2+bx+c\) to get three simultaneous equations, then solve them (usually with your GDC's equation solver). If you have several data points, use quadratic regression instead.
Do I need calculus to find the maximum or minimum?
No - at AI SL you find the vertex using the formula \(x=-b/(2a)\), or read it straight off your GDC's graph using the maximum/minimum tool. Both are accepted methods and the GDC approach is usually faster.
Why do these questions ask for a "reasonable domain"?
A quadratic model is only trustworthy over the range where it was built to apply - for example, time can't be negative, and a fencing width can't exceed half the total fencing. Stating a sensible domain shows you understand the model is a simplification of a real situation, not an equation that holds for all x.
Sub-topics
Quadratic Models broken down into its individual skills, each with its own focused page.
Related topics
More Functions & Modelling topics from the same AI SL syllabus unit, in case you want to keep going.