GraphsStudy Notes

Worked Examples

Part of Reciprocal Graphs — GCSE Mathematics

This study notes covers Worked Examples within Reciprocal Graphs for GCSE Mathematics. Revise Reciprocal Graphs in Graphs for GCSE Mathematics with 11 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 9 of 11 in this topic. Use this study notes to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 9 of 11

Practice

11 questions

Recall

10 flashcards

✏️ Worked Examples

Example 1: Completing a Table of Values for y = 6/x

Question: Complete the table of values for y = 6/x for x = -6, -3, -2, -1, 1, 2, 3, 6. Describe the two branches of the graph.

Show Solution

Calculate y for each x value:

x = -6: y = 6/(-6) = -1 x = -3: y = 6/(-3) = -2 x = -2: y = 6/(-2) = -3

x = -1: y = 6/(-1) = -6 x = 1: y = 6/1 = 6 x = 2: y = 6/2 = 3

x = 3: y = 6/3 = 2 x = 6: y = 6/6 = 1

Note: x = 0 is NOT included — division by zero is undefined.

Branches: k = 6 > 0, so branches lie in Quadrant 1 (positive x, positive y) and Quadrant 3 (negative x, negative y). Both approach but never touch the axes.

Answer: y values: -1, -2, -3, -6, 6, 3, 2, 1. Hyperbola with branches in Q1 and Q3.

Example 2: Finding the Equation from a Point on the Curve

Question: A reciprocal graph y = k/x passes through the point (4, 3). Find k and write the equation of the curve.

Show Solution

Step 1: Use the property xy = k — k = 4 × 3 = 12

Step 2: Write the equation — y = 12/x

Check: at x = 6, y = 12/6 = 2. Product: 6 × 2 = 12 ✓

Answer: k = 12, equation: y = 12/x

Practice Questions for Reciprocal Graphs

The graph of y = 1/x has an asymptote along the x-axis. What does this mean?

A. The graph touches the x-axis at x = 0
B. The graph crosses the x-axis at x = 1
C. The graph gets closer and closer to the x-axis but never reaches it
D. The graph is a straight line along the x-axis

1 markfoundation

Explain why the graph y = 5/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

2 markshigher

Quick Recall Flashcards

What does the graph of y = 1/x look like?

A hyperbola — two separate curved branches. - One branch in the top-right (positive x, positive y) - One branch in the bottom-left (negative x, negative y) The curve gets closer and closer to both axes but never touches them.

What is an asymptote and where are they on y = k/x?

An asymptote is a line the curve approaches but never reaches or crosses. For y = k/x: - Vertical asymptote: x = 0 (the y-axis) - The function is UNDEFINED when x = 0 - Horizontal asymptote: y = 0 (the x-axis) - y never equals zero for any finite x