Maths — Pure 1

Can we identify the right sketch graph of this reciprocal function?

Review questionbeyond the specification map

Beyond the specification map. NON-OVERLAP for this lane: reciprocal-graph sketch via completing the square — rides 1.1.1/quadratics and curve sketching; the range work exceeds P1 1.2 "simple cases". It is kept in the library for extra maths practice, not as specification-point coverage.

Can we identify the right sketch graph of this reciprocal function?

Question

Which of the graphs below is a sketch of $y = \frac{1}{4x-x^2-5}\; ?$

Suggestion

Would it help to complete the square for the quadratic expression in the denominator?

Solution

We can complete the square. We have $4x-x^2-5 = -(x^2-4x) -5 = -((x-2)^2-4)-5 = -(x-2)^2+4-5 = -(x-2)^2-1$ .

So the graph is $y = \frac{1}{-(x-2)^2-1},$ which shows the function is always negative, and has a minimum value of $y=-1$ at $x=2$ .

Hence the correct graph is c.

Graph that tends to zero as x tends to plus or minus infinity, with a maximum at a positive x value, and y positive for all x.

Graph that tends to zero as x tends to plus or minus infinity, with a maximum at a negative x value, and y positive for all x.

Graph that tends to zero as x tends to plus or minus infinity, with a minimum at a positive x value, and y negative for all x.

Graph that tends to zero as x tends to plus or minus infinity, with a minimum at a negative x value, and y negative for all x.