Unit 7

5 marks · Easy

The function models the height (in metres) of a walkway at position metres, for .

(a)

Using the graph of , state the values of for which the walkway is below ground level (i.e. where ).

[2 marks]

(b)

Solve the inequality . Write your answer as a set of values of .

[3 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a) The walkway is below ground when and (i.e. all x between roots apart from x = 0). · must show identification of below axis region, gives x-intervals where y<0

(b) and · must show correct solution intervals, clear inequality notation

Common mistakes: Only giving points rather than intervals; Not solving inequality by sign analysis

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Identifies that below ground means or looks for negative y values on the graph.

A1 (a) · 1 mark · States the two correct intervals of where : and .

M2 (b) · 1 mark · Solves and finds roots: .

M3 (b) · 1 mark · Tests sign in intervals or sketches sign diagram/graph.

A2 (b) · 1 mark · Gives the combined solution set: and .

Worked solution

a) To find where the walkway is below ground, we need .

The graph of crosses the x-axis at .

Between and , the value of is negative.

Between and , again is negative.

So, the walkway is below ground when and .

b) To solve :

First, solve :

So roots at .

Checking signs between roots (using test values for x):

For , e.g. : (negative)

For , e.g. : (positive)

For , e.g. : (negative)

For , e.g. : (positive)

But above, b) wants , so solution is where negative.

So the solution is and .