Optimisation of a Piecewise Function

00:00 5 marks

Consider the piecewise function defined as follows:
\[f(x) = \begin{cases} -x^2 + 4x + 1 & \text{for } x < 2 \\ -2x + 5 & \text{for } x \geq 2 \end{cases}\]

Answer the following parts using extended GDC techniques.

Part a

3 minutes 2 marks

Question

Using a graphing calculator (GDC), plot the piecewise function. Sketch the graph in your answer booklet, ensuring key features are clearly labelled.

Identify and list the coordinates of three key points on each interval that illustrate the function’s behavior.

Hint Hint

Hint

For the quadratic part, consider the vertex, y-intercept, and endpoint. For the linear part, consider the starting point and two other points to show the line’s direction.

Answer Answer
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Complete Answer

Key points for quadratic part (\(x < 2\)): Vertex at \((2, 5)\) (endpoint, not included), y-intercept \((0, 1)\), point at \((1, 4)\). Key points for linear part (\(x \geq 2\)): Starting point \((2, 1)\), and points \((3, -1)\), \((4, -3)\).

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Part b

4 minutes 3 marks

Question

Using numerical methods on your GDC, determine the coordinates of the local maximum of the overall piecewise function. Show your GDC screen or describe the method used.

Hint Hint

Hint

Examine the graph from part (a) to identify the interval where the local maximum is likely to occur. Use GDC’s ‘maximum’ function or table feature to refine your search.

Answer Answer
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Complete Answer

Local maximum is at \((2, 5)\).

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