Grade 10 · Grade 10: Polynomials, Models and Function Behaviour · Polynomials, Models and Function Behaviour
Quadratic and Exponential Models
Build models from data, convert between quadratic forms and solve exponential equations with logarithms.
Learning objectives
- Build quadratic models and convert between standard, factored and vertex forms.
- Interpret model parameters in context.
- Graph exponential growth and decay.
- Use logarithms to solve simple exponential equations.
AERO Mathematics alignment
AERO.M10.AE.4
Build quadratic models from data or descriptions, convert between standard, factored and vertex forms, and interpret parameters in context.
AERO.M10.AE.5
Graph exponential functions, describe growth and decay, and use logarithms to solve simple exponential equations.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A stadium roof arch and a decaying medicine dose both make curves, but only one keeps returning to zero and rising again.
Think about it
Which situation is quadratic and which is exponential, and how would you tell from a table?
Hint: Look for constant second differences against constant ratios.
EXPLAIN
Building and reading models
Three forms of a quadratic
| Form | Equation | Reveals |
|---|---|---|
| Standard | y = ax² + bx + c | The y-intercept c |
| Factored | y = a(x − p)(x − q) | The zeros p and q |
| Vertex | y = a(x − h)² + k | The vertex (h, k) |
Complete the square to move from standard to vertex form: y = x² − 8x + 11 becomes y = (x − 4)² − 5, so the minimum is −5 at x = 4.
Exponential models
y = a · bˣ grows when b > 1 and decays when 0 < b < 1. Doubling time and half-life describe how quickly this happens. Radioactive decay, cooling, medicine in the bloodstream and compound interest all follow this form.
Logarithms
A logarithm answers the question what exponent do I need. If 3ˣ = 81, then x = log₃81 = 4. Logs turn an unknown exponent into an ordinary unknown you can solve for.
Remember
Quadratics have a turning point; exponentials do not. Exponential decay approaches zero but never reaches it.
INVESTIGATION
Try it yourself
Measure a bouncing ball's rebound height for five bounces. Decide whether the pattern is exponential, find the ratio, write the model and predict bounce eight.
Watch
Solving Quadratic Equations Using the Quadratic Formula
The Organic Chemistry Tutor · 5:55
You will need fluent quadratic solving to interpret your models.
Before you watch: How does vertex form show the maximum immediately?
- Convert y = x² − 8x + 11 to vertex form.
- Solve 3^x = 81 using logarithms.
Interactive simulation · PhET
Graphing Quadratics
Match a parabola to a projectile path, then read the maximum from vertex form.
While you explore
- Where is the maximum height, and how does vertex form show it?
- What do the x-intercepts mean in a projectile context?
Key vocabulary
- Logarithm
- The exponent you need to reach a number.
- Vertex form
- A way of writing a quadratic that shows its turning point.
Practice questions
0/1 correct
Level 1 · Criterion A
Solve 2ˣ = 32.
MYP criterion tasks
Level 2 · Criterion D
A ball's height is h = −5t² + 30t. Find the maximum height and the time it occurs.
Reflect & track
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