Understand the gap in a reciprocal graph

By UK GCSE Mathematics · independent study · myAudience

Original myAudience mathematics teaching illustration; labels, relationships and lesson calculations independently checked. · Original project artwork; no exam-board figures or textbook illustrations reused.

Understand the gap in a reciprocal graph

The rule y = 1/x is undefined at x = 0 because division by zero is not permitted. Positive inputs give positive outputs, and negative inputs give negative outputs. Its two branches lie in the first and third quadrants.

As positive x becomes large, 1/x approaches zero while remaining positive. As positive x approaches zero, 1/x becomes arbitrarily large. The coordinate axes are asymptotes: the curve approaches them but never crosses them.

Do not join plotted points across the missing input zero. That would invent outputs the rule does not have. The domain restriction is part of the function, not a minor detail that can be removed to make a tidy picture.

Worked example: Evaluate 1/x at x = 2, 1/2 and −2.

Divide 1 by each nonzero input.

Keep the sign of a negative divisor.

Answer: 1/2, 2 and −1/2.

Check your understanding: Does the graph pass through the origin?

Answer: No.

Why: At x = 0 the expression is undefined.