Quadratic Relationships

Unlike linear models, which have a constant rate of change, quadratic models are used when the dependent variable increases and then decreases, or decreases and then increases.

1. The Structure of a Quadratic Model

• The formula for a quadratic relationship is written in the form $y = ax^2 + bx + c$, where the most defining feature is the presence of a squared term ($x^2$).
• The graph of a quadratic model is a symmetrical curve called a parabola.
• A quadratic graph features a single turning point (either a maximum or minimum), a y-intercept, and sometimes x-intercepts (where the curve crosses the horizontal axis).

2. The Shape of the Parabola

The overall shape of the graph is dictated entirely by the coefficient of the $x^2$ term:

• Positive $x^2$: If the squared term is positive (e.g., $2x^2$), the graph forms a 'u' shape with a minimum turning point at the bottom.
• Negative $x^2$: If the squared term is negative (e.g., $-2x^2$), the graph forms an 'n' shape with a maximum turning point at the top.

3. Evaluating and Real-World Validity

• Evaluating simply means substituting a given input value into your formula to calculate the output.
• When evaluating real-world models, you must always consider the validity of your answers. Mathematical formulas will happily generate negative numbers indefinitely, but in the real world, things like physical height, time, or the number of objects cannot drop below zero.

4. Quadratic Models in Spreadsheets

• To explore a quadratic model in Excel, you should generate a sequence of $x$ values that includes both negative and positive numbers to reveal the true shape of the curve.
• When typing your formula into the spreadsheet, you must use the ^ symbol to indicate a power (e.g., ^2 for squared) and the * symbol for multiplication.