Further Modelling with Functions

Linear Models ($y = mx + c$)

• Interpretation: The gradient $m$ represents a constant rate of change (e.g., growth per month), while the y-intercept $c$ represents the initial value at $t=0$.
• Application: Used when a quantity increases or decreases by the same fixed amount over every interval.

Quadratic Models ($y = ax^2 + bx + c$)

• Interpretation: Often used for projectile motion where $y$ is height and $x$ is time. The coefficient $a$ is usually negative, creating a downward parabola.
• Key Features: The maximum or minimum point (vertex) is found by completing the square or using $x = -\frac{b}{2a}$. The roots of the equation represent when the object hits the ground ($y=0$).

• Initial Value vs. Rate: In a linear model, the constant term is the starting point. In a quadratic model, the constant term is also the starting point ($t=0$), but the 'rate' is not a single number; it is a function of the independent variable.

• Domain Restrictions: Models often only work for specific ranges (e.g., $t \ge 0$). It is vital to distinguish between the mathematical domain of a function and the practical domain of the real-world scenario.

• Contextualize the Constants: Always ask what the numbers in the formula represent in the real world. If a formula is $H = 20 + 5t - t^2$, the $20$ is the starting height, not just a 'y-intercept'.

• Check for Validity: If a question asks for a criticism, look for 'extreme values'. Does the model predict a negative height? Does it suggest a population will grow forever? These are standard points for losing marks if ignored.

• Algebraic Precision: When dealing with quadratic models, completing the square is the most reliable way to find the maximum height and the time it occurs. Always show the step-by-step transformation to the form $a(x-h)^2 + k$.

• Sanity Checks: Ensure your answers make sense. A stone thrown from a cliff should not take 500 seconds to hit the water, nor should a factory produce a negative number of items.

• Confusing Gradient with Total: In linear models, students often confuse the rate of change (gradient) with the total value at a specific time. Always check if the question asks for the 'increase' or the 'total amount'.

• Ignoring Units: Modelling questions are grounded in reality. Forgetting to include units (meters, seconds, dollars) or failing to convert units (minutes to hours) is a frequent source of error.

• Misinterpreting Roots: In quadratic projectile models, solving $h=0$ often gives two values for $t$. Students must realize that a negative time value is mathematically valid but contextually impossible and must be rejected.