Quadratic Models, Applying and Interpreting

Every feature of a parabola has a specific real-world meaning that depends entirely on what the variables represent. The interpretation is never just the coordinates, it is the story those coordinates tell.

What each feature tells you:

• $y$-intercept: The value of the modelled quantity when $x = 0$, the initial or starting value. E.g. if $x$ = time, this is the quantity at time zero. If $x$ = units produced, this is the fixed cost with zero production.
• $x$-intercepts: Where the quantity is zero. E.g. when height = 0 (ball at ground), when profit = 0 (break-even), when area = 0 (no paddock). There may be two $x$-intercepts bounding the practical range.
• Vertex: The maximum (opens down) or minimum (opens up). Always the most important point: maximum height reached, maximum area enclosed, minimum cost, etc. State both the $x$ and $y$ coordinates with units and meaning.

The y-intercept gives the initial value (when x = 0); the x-intercepts give the zeros of the model; the vertex gives the maximum or minimum output value and the input at which it occurs.

Pause, copy the three contextual meanings: y-intercept = initial value when x = 0, x-intercepts = input values where the output is zero, vertex = maximum or minimum output value, with a real example for each.

The y-intercept gives the starting value, x-intercepts give the zeros, and the vertex gives the optimal value in context, but a parabola extends infinitely in both directions, while real situations do not. The valid domain limits the model: time can't be negative, a ball can't be below the ground, and a length can't exceed the available material.

A mathematical model produces output for any $x$-value, but only a restricted range gives physically meaningful results. Identifying the valid domain is a critical thinking skill, NESA tests it explicitly.

How to determine the valid domain:

1. Identify the physical quantity on the $x$-axis (time, width, units produced, etc.)
2. Determine what values are physically possible negative time is impossible; a paddock width can't exceed the total fence; a price can't be negative.
3. Use the $x$-intercepts to find where the modelled quantity ($y$) becomes zero, usually a natural boundary.
4. State the domain as an inequality with its meaning: e.g. "$0 \leq w \leq 40$ where $w$ is the width in metres, width must be non-negative and at most 40 m."

Model limitations, what to say: NESA expects specific, contextual limitations. Good responses name an assumption the model makes and explain why that assumption might not hold.

• "The model assumes the shape is exactly a parabola, but the projectile may be affected by wind resistance, changing the curve."
• "The model predicts positive area for any width between 0 and 40 m, but practical constraints (e.g. minimum width for animals) would further restrict the domain."
• "The profit model assumes a fixed price per unit, but in reality prices fall as supply increases."

A model's valid domain is the set of x-values that make physical sense in context, for example, time cannot be negative and a ball cannot be below the ground. Always state domain restrictions explicitly in HSC answers.

Pause, copy the valid domain definition (set of x-values that make physical sense) and the two common restriction types: x ≥ 0 for time/quantity, and an upper bound from the physical context into your book.

Once the valid domain is established, every exam question about a quadratic model asks for one of three things: the vertex (maximum height or minimum cost), an intercept (when a quantity starts or reaches zero), or a y-value at a specified x. Each technique involves reading the graph directly, always include units in your answer.

NESA provides you with a graph and asks you to answer questions about it, you must read values accurately, identify features, and translate numbers into contextual answers.

What NESA questions typically ask:

• "What is the maximum [height/area/profit]?" → Read the $y$-coordinate of the vertex.
• "When does [the ball land / the quantity return to zero]?" → Read the positive $x$-intercept.
• "What is the [value/height/cost] at $t = k$?" → Read the $y$-value at the specified $x$.
• "For what values of $x$ is [the quantity] greater than $k$?" → Find where the graph is above $y = k$ (read the two $x$-values where the graph meets the horizontal line $y = k$).
• "State the valid values of $x$ for this model." → State the domain with a brief justification.

To solve practical problems from a quadratic graph: read the required feature directly (vertex for max/min, intercept for start/end, or a specific y-value by reading across then down). Always state units in your answer.

Pause, copy the three read-off techniques: read the vertex for max/min, read an intercept for start/end, read horizontally then vertically for a given x-value, and always state the unit of each answer.