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Quadratic Functions Modelling

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Quadratic Functions Modelling

Use quadratic functions to investigate profit, motion and design. Show substitutions and calculations where asked. Give units and interpret answers in context. Assume each model is valid only within the stated practical domain.

Read the model

A community sports club is testing quadratic models for several projects.

1.A ball’s height above the ground is modelled by h(t) = −5t² + 20t, where h is in metres and t is in seconds. Find the time when it reaches its maximum height, then find that height.

2.For the ball model h(t) = −5t² + 20t, solve h(t) = 0. Which solution gives the time it lands after being kicked, and why is the other solution also reasonable?

3.A shop models its daily profit by P(x) = −2x² + 120x − 1000, where x is the number of items sold and P is in dollars. How many items give the maximum modelled profit, and what is that profit?

4.For P(x) = −2x² + 120x − 1000, find the break-even sales levels by solving P(x) = 0. Round to the nearest whole item and state what the roots mean.

5.Select the statement that best explains the coefficient a = −2 in P(x) = −2x² + 120x − 1000.

The model opens upward, so it has a minimum profit.
The model opens downward, so it has a maximum at its vertex.
The starting profit is −$2.
Profit drops by exactly $2 for every item sold.

Build and test a model

Use variables, graphs and evidence to justify conclusions.

6.A rectangular garden is built against a straight wall, so fencing is needed for only three sides. There are 48 m of fencing. If each perpendicular width is x metres, write the length L(x) and area A(x) in terms of x.

7.For the garden model A(x) = −2x² + 48x, find the dimensions that give the greatest area and calculate that area.

8.A model for a ball’s height is h(t) = −5t² + 20t. Complete the table, then use the values to identify the axis of symmetry and explain how the table supports it.

t (s)

h(t) (m)

___

9.A venue’s ticket income model is R(n) = −3n² + 180n, where n is the ticket price in dollars and R is daily income in dollars. Find the ticket price that maximises income and the maximum income. State one limitation of using this model alone to set a real ticket price.

10.A student claims that in y = ax² + bx + c, changing c changes the width of the parabola. Is the claim correct? Explain what c changes and which coefficient controls the vertical stretch or compression.

11.A quadratic model predicts that a business’s profit is positive between its two break-even sales levels. Explain why the vertex gives the greatest profit when the quadratic coefficient is negative.