Analyse and optimise a quadratic model

A programmed drone flight is modelled by the rule below. What is the greatest modelled height?

Change the end of the allowed time. The curve shows only the allowed interval, while “turn” marks the algebraic turning point. The dashed horizontal line is h=8.

What happens as the allowed end time passes 3 seconds?

Read a turning-point form

In the displayed form, the square is zero at the turning-point input, leaving the constant output. A positive a gives a minimum; a negative a gives a maximum. The sign inside the bracket is reversed when finding h.

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

For h(t)=−(t−3)²+12, at what time is the square zero?

Make the expression inside the square equal to zero

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Time

\(\displaystyle t=3\)

Find the height at that time

Substitute the turning-point time into the height rule

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Why is this turning point a maximum?

The domain can exclude the turning point

If the flight record ends at t=2, the time t=3 is not allowed. Compare the allowed endpoints or use the increasing graph. The greatest recorded model height is h(2)=11, not 12.

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Domain

\(\displaystyle 0\le t\le2\)

What is the greatest modelled height on 0≤t≤2?

Check whether the turning point is allowed; otherwise compare endpoints

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Domain

\(\displaystyle 0\le t\le2\)

A report uses 12 m as the maximum. What is the error?

Minimum models need the same domain check

For y=(t−4)²+2 on 0≤t≤3, the unrestricted minimum occurs at t=4. It is outside the interval, so compare the endpoints: y(0)=18 and y(3)=3.

Rule

\(\displaystyle y=(t-4)^2+2\)

Domain

\(\displaystyle 0\le t\le3\)

What is the smallest allowed output for that model?

Check the domain before using the unrestricted minimum

Find when a specified level is reached

For the drone model, set h=8 and solve both branches. The graph meets the horizontal line h=8 at the same two times

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Height

\(\displaystyle h=8\)

Domain

\(\displaystyle 0\le t\le6\)

On 0≤t≤6, what is the earlier time at height 8 m?

Solve both square-root branches and choose the smaller allowed time

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Height

\(\displaystyle h=8\)

Domain

\(\displaystyle 0\le t\le6\)

What is the later time?

Solve both branches and choose the larger allowed time

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Domain

\(\displaystyle 0\le t\le4\)

Which time at height 8 m lies in the allowed interval?

Rule

\(\displaystyle h(t)=-(t-3)^2+12\)

Domain

\(\displaystyle 0\le t\le6\)

How many times does the model reach height 13 m?

Use graphs and numbers alongside algebra

A threshold can have zero, one or two allowed crossings. Read approximate intersections from a graph and substitute them to check. Later graph lessons develop numerical approximation when roots are not simple exact values

Compare the model with observations

At t=2 the rule predicts 11 m. An observation is 10.6 m. The absolute discrepancy is 0.4 m. Check whether this is reasonable for the measurement precision and intended use; the model is an approximation of a programmed flight

Prediction

\(\displaystyle 11\text{ m}\)

Observation

\(\displaystyle 10.6\text{ m}\)

Calculate the absolute difference between 11 m and 10.6 m.

Subtract the observed value from the prediction, then take the absolute value

At 2 s, a drone model predicts 11 m and the observed height is 10.6 m. Which conclusion is justified?

Communicate the result

Report the allowed interval, turning point, endpoint checks, threshold solutions with units, and any excluded roots. Include a labelled graph or table, a comparison with observations and a limitation. Do not extend a flight or area model beyond its stated domain without new evidence