Reciprocal Functions and Asymptotes

Understanding graphs of y = 1/x, y = 1/x², and rational functions y = (ax + b)/(cx + d)

SPEC 2.7 • ALGEBRA AND FUNCTIONS

The Graph of y = 1/x

Key Features

Shape: Rectangular hyperbola

Branches in Quadrants 1 and 3

Passes through (1, 1) and (-1, -1)

Rotational symmetry order 2 about origin

Asymptotes

Vertical: x = 0 Horizontal: y = 0

The curve approaches but never touches these lines.

The Graph of y = 1/x²

Key Features

Both branches above the x-axis

Branches in Quadrants 1 and 2

Passes through (1, 1) and (-1, 1)

Reflection symmetry in y-axis

Why Always Positive?

Since x² is always positive (for x ≠ 0), then 1/x² is also always positive. The curve never goes below the x-axis.

Variations: y = a/x

a > 0: Branches in Q1 and Q3

a < 0: Branches in Q2 and Q4

Key Points

y = a/x passes through (1, a) and (-1, -a). Larger |a| moves the curve further from the axes. Asymptotes always x = 0, y = 0.

Variations: y = a/x²

a > 0: Both branches above x-axis

a < 0: Both branches below x-axis

Key Points

y = a/x² passes through (1, a) and (-1, a). Always symmetric in the y-axis. Asymptotes always x = 0, y = 0.

Translations: y = 1/(x + a) + b

Translation Rules

y = \frac{1}{x + a} + b

"+a" inside: Shifts graph LEFT by a

New vertical asymptote: x = -a"+b" outside: Shifts graph UP by b

New horizontal asymptote: y = b

Example: y = 1/(x - 2) + 3

Translation of y = 1/x by vector

(23)

Vertical asymptote: x = 2 Horizontal asymptote: y = 3

Common Mistake

y = 1/(x + 2) shifts LEFT by 2, not right! The shift is opposite to the sign inside.

Rational Functions: y = (ax + b)/(cx + d)

Functions like

y = \frac{3 x}{x - 1}

,

y = \frac{2 x + 1}{x - 3}

are rational functions.

Vertical Asymptote

Set denominator = 0

c x + d = 0 ⟹ x = - \frac{d}{c}

Horizontal Asymptote

Ratio of leading coefficients

y = \frac{a}{c}

Finding Intercepts

y-intercept: Set x = 0, calculate y = b/d

x-intercept: Set numerator = 0, solve ax + b = 0

Worked Example01

Sketch the graph of $y = \frac{1}{x - 2} + 3$ , stating the equations of the asymptotes and coordinates of any axis intercepts.

1

Vertical asymptote: x - 2 = 0 → x = 2

2

Horizontal asymptote: y → 3 as x → ±∞ → y = 3

3

y-intercept: x = 0: y = 1/(0-2) + 3 = -1/2 + 3 = 5/2 → (0, 5/2)

4

x-intercept: 0 = 1/(x-2) + 3 → 1/(x-2) = -3 → x - 2 = -1/3 → x = 5/3

Intercepts: (0, 5/2) and (5/3, 0)

Worked Example02

Sketch the graph of $y = \frac{3 x}{x - 1}$ , finding the asymptotes and any axis intercepts.

1

Vertical asymptote: x - 1 = 0 → x = 1

2

Horizontal asymptote: y = 3/1 = y = 3

3

y-intercept: x = 0: y = 0/(0-1) = 0 → (0, 0)

4

x-intercept: 3x = 0 → x = 0 → (0, 0)

5

Curve passes through origin with asymptotes x = 1, y = 3

Worked Example03

Sketch the graph of $y = \frac{2 x + 1}{x - 2}$ , stating asymptotes, intercepts, domain and range.

1

Vertical asymptote: x - 2 = 0 → x = 2

2

Horizontal asymptote: y = 2/1 = y = 2

3

y-intercept: x = 0: y = 1/(-2) = -1/2 → (0, -1/2)

4

x-intercept: 2x + 1 = 0 → x = -1/2 → (-1/2, 0)

5

Domain: x ∈ ℝ, x ≠ 2 Range: y ∈ ℝ, y ≠ 2

Worked Example04

Show that $y = \frac{3 x - 5}{x - 2}$ can be written as $y = 3 + \frac{1}{x - 2}$ , and hence sketch the curve.

1

Write 3x - 5 = A(x - 2) + B. Expanding: Ax - 2A + B

2

Comparing coefficients: A = 3, and -2A + B = -5 → -6 + B = -5 → B = 1

3

So 3x - 5 = 3(x - 2) + 1

4

y = \frac{3 ( x - 2 ) + 1}{x - 2} = 3 + \frac{1}{x - 2}

✓

5

This is y = 1/x translated by (2, 3). Asymptotes: x = 2, y = 3

Worked Example05

Find the coordinates where $y = \frac{x + 2}{x - 1}$ intersects the line y = 2x.

1

Set equal:

\frac{x + 2}{x - 1} = 2 x

2

Multiply by (x - 1): x + 2 = 2x(x - 1) = 2x² - 2x

3

Rearrange: 2x² - 3x - 2 = 0

4

Factorise: (2x + 1)(x - 2) = 0 → x = -1/2 or x = 2

5

Using y = 2x: When x = -1/2, y = -1. When x = 2, y = 4

Intersection points: (-1/2, -1) and (2, 4)

Worked Example06

Sketch $y = \frac{1}{( x - 1 ) ^{2}} + 2$ , stating the asymptotes and any axis intercepts.

1

Vertical asymptote: (x-1)² = 0 → x = 1

2

Horizontal asymptote: As x → ±∞, y → 2 → y = 2

3

y-intercept: x = 0: y = 1/(0-1)² + 2 = 1 + 2 = 3 → (0, 3)

4

x-intercept: 0 = 1/(x-1)² + 2 → 1/(x-1)² = -2. No solution!

5

Shape like y = 1/x² translated right 1, up 2. Both branches above y = 2.

No x-intercept (curve always above y = 2)

Practice

Question 1: Sketch the graph of

y = \frac{1}{x + 3} - 2

, stating the equations of the asymptotes and the coordinates of any axis intercepts.

Vertical asymptote: x + 3 = 0 → x = -3

Horizontal asymptote:y = -2

y-intercept: x = 0: y = 1/3 - 2 = -5/3 → (0, -5/3)

x-intercept: 0 = 1/(x+3) - 2 → 1/(x+3) = 2 → x + 3 = 1/2 → x = -5/2

Practice

Question 2: Sketch the graph of

y = \frac{4 x - 1}{x + 2}

, stating the asymptotes and axis intercepts.

Vertical asymptote: x + 2 = 0 → x = -2

Horizontal asymptote: y = 4/1 = y = 4

y-intercept: x = 0: y = -1/2 → (0, -1/2)

x-intercept: 4x - 1 = 0 → x = 1/4 → (1/4, 0)

Practice

Question 3: Find the intersection points of

y = \frac{2 x}{x - 3}

and y = x + 1.

Step 1: Set equal:

\frac{2 x}{x - 3} = x + 1

Step 2: Multiply by (x - 3): 2x = (x + 1)(x - 3) = x² - 2x - 3

Step 3: Rearrange: x² - 4x - 3 = 0

Step 4: Quadratic formula: x = (4 ± √28)/2 = 2 ± √7

Step 5: y = x + 1: y = 3 ± √7

Intersection points: (2 + √7, 3 + √7) and (2 - √7, 3 - √7)

Summary

Key Results

FunctionVertical AsymptoteHorizontal Asymptote

y = 1/x, y = 1/x²x = 0y = 0

y = 1/(x + a) + bx = -ay = b

y = (ax + b)/(cx + d)x = -d/cy = a/c

Key Exam Skills

Vertical asymptote: set denominator = 0

Horizontal asymptote: ratio of leading coefficients

y-intercept: set x = 0; x-intercept: set numerator = 0

Domain excludes vertical asymptote; range excludes horizontal asymptote