Quadratic Relationships

The height $\displaystyle h$, in metres, of an object after $\displaystyle t$ seconds is modelled by $\displaystyle h = 40t - 5t^2$.
Calculate the height after $\displaystyle 3$ seconds.

The height $\displaystyle h$, in metres, of an object after $\displaystyle t$ seconds is modelled by $\displaystyle h = 40t - 5t^2$.
Calculate the height after $\displaystyle 8$ seconds and interpret your answer.

A quadratic model is $\displaystyle y = 12 + 5x - x^2$.
Calculate $\displaystyle y$ when $\displaystyle x = 4$.

A spreadsheet is used to model $\displaystyle y = 15 + 4x - 2x^2$.
Column A holds the $\displaystyle x$ values, with the first value in cell A2.
Write down the formula to enter in cell B2 to calculate $\displaystyle y$.

A distress flare is fired vertically.
Its height $\displaystyle h$, in metres, after $\displaystyle t$ seconds is modelled by $\displaystyle h = 60t - 5t^2$.
Calculate the height of the flare after $\displaystyle 10$ seconds.

A spreadsheet calculates the height $\displaystyle h = 60t - 5t^2$ of a flare at each whole second.
Part of the table is shown.
$\displaystyle t = 4, h = 160$; $\displaystyle t = 5, h = 175$; $\displaystyle t = 6, h = 180$; $\displaystyle t = 7, h = 175$; $\displaystyle t = 8, h = 160$
State the greatest height shown in the table, and the time at which it occurs.

A company models its weekly profit $\displaystyle P$, in pounds, against the selling price $\displaystyle p$, in pounds, by $\displaystyle P = -2p^2 + 60p - 250$.
Calculate the profit when the selling price is $\displaystyle \pounds 10$.

A company models its weekly profit by $\displaystyle P = -2p^2 + 60p - 250$, where $\displaystyle p$ is the selling price in pounds.
Determine whether a price of $\displaystyle \pounds 12$ or a price of $\displaystyle \pounds 16$ gives the greater profit. Use your working to justify your answer.

A company sets the price of a new product.
If the price is very low it sells many units but makes very little on each one.
If the price is very high it makes a lot on each unit but sells almost none.
Explain why a quadratic model is a sensible shape for the profit.

The height of a ball is modelled by $\displaystyle h = 25 + 10t - 5t^2$, where $\displaystyle h$ is in metres and $\displaystyle t$ is in seconds.
Explain why this model is not valid when $\displaystyle t = 7$.