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The diagram below shows the graph of a cubic function $y = g(x)$, with stationary points at $x = -2$ and $x = 4$ - Scottish Highers Maths - Question 5 - 2019

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Question 5

The-diagram-below-shows-the-graph-of-a-cubic-function-$y-=-g(x)$,-with-stationary-points-at-$x-=--2$-and-$x-=-4$-Scottish Highers Maths-Question 5-2019.png

The diagram below shows the graph of a cubic function $y = g(x)$, with stationary points at $x = -2$ and $x = 4$. On the diagram in your answer booklet, sketch the... show full transcript

Worked Solution & Example Answer:The diagram below shows the graph of a cubic function $y = g(x)$, with stationary points at $x = -2$ and $x = 4$ - Scottish Highers Maths - Question 5 - 2019

Step 1

Identify shape and roots

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Answer

The graph of the cubic function has roots at the stationary points x=−2x = -2 and x=4x = 4. Since these points are where the derivative function changes sign, they correspond to local maxima and minima. The graph of the derivative y=g′(x)y = g'(x) will reflect the behavior of the cubic function around these roots.

Step 2

Interpret shape

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Answer

The graph of y=g′(x)y = g'(x) should have a parabolic shape opening upwards. Since there is a local maximum at x=−2x = -2 and a local minimum at x=4x = 4, the derivative will be positive between these points. The turning point of the parabola (the vertex) will be around x=1x = 1. Therefore, the sketch should show an upward-opening parabola intersecting the x-axis at x=−2x = -2 and x=4x = 4, and a minimum turning point at x=1x = 1.

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