Further Pure Mathematics 1 (Paper 1): Topic 1.2 Rational Functions and Graphs

Hello and welcome to this critical section of Further Pure Mathematics! Graphing rational functions isn't just about drawing pretty curves; it’s about understanding the limits, extremes, and prohibited values of complex equations. This skill is vital for solving advanced inequalities and interpreting mathematical models. Don't worry if these graphs look intimidating—we’ll break them down into simple, manageable steps.

1. Defining Rational Functions and Their Structure

What is a Rational Function?

A Rational Function \( R(x) \) is essentially a fraction where both the numerator and the denominator are polynomials. It has the general form:

\[ R(x) = \frac{P(x)}{Q(x)} \]

where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \).

In this syllabus (9231), we typically deal with "simple" rational functions where the degree of the numerator and the denominator are at most 2.
Example: \( y = \frac{x^2 + 2x}{x-3} \) or \( y = \frac{x^2 - 1}{x^2 + 4} \).

Key Takeaway:

Rational functions are simply polynomial division, but the points where the denominator is zero (the function's "no-entry zones") create the most interesting features—the asymptotes.

2. The Invisible Barriers: Asymptotes

Asymptotes are lines (or curves) that the graph approaches but never touches (or only touches at infinity). They are the most crucial elements when sketching rational functions.

2.1 Vertical Asymptotes (V.A.)

These occur where the denominator \( Q(x) \) is zero, provided there isn't a common factor causing a hole in the graph instead.

  • How to find: Set the denominator \( Q(x) = 0 \) and solve for \( x \).
  • Equation form: \( x = a \) (a vertical line).

Example: For \( y = \frac{x+1}{x-2} \), the V.A. is \( x=2 \).

Common Mistake to Avoid: If \( P(x) \) and \( Q(x) \) share a factor (e.g., \( y = \frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} \)), there is a hole (or removable singularity) at \( x=2 \), not a V.A. Always simplify the function first!

2.2 Horizontal Asymptotes (H.A.)

These describe the behaviour of the function as \( x \to \pm \infty \).

  • Compare Degrees: Let \( n \) be the degree of \( P(x) \) and \( m \) be the degree of \( Q(x) \).
  • Case 1: \( n < m \) (Numerator degree is smaller)

    The H.A. is always the x-axis: \( y = 0 \).

    Analogy: If you divide a small number by an infinitely large number, the result approaches zero.

  • Case 2: \( n = m \) (Degrees are equal, max degree 2)

    The H.A. is the ratio of the leading coefficients: \( y = \frac{a_n}{b_m} \).

    Example: For \( y = \frac{3x^2 + 5}{x^2 - 1} \), the H.A. is \( y = \frac{3}{1} = 3 \).

  • Case 3: \( n > m \)

    No H.A. The function tends towards \( \pm \infty \). This leads to an Oblique Asymptote if \( n = m+1 \).

2.3 Oblique (Slant) Asymptotes (O.A.)

This is a key focus area for Further Maths. An O.A. exists when the degree of the numerator is exactly one greater than the degree of the denominator (i.e., \( n = m+1 \)). Since the degrees must be at most 2, we are focused on \( n=2 \) and \( m=1 \) cases (e.g., quadratics divided by linears).

Step-by-Step: Finding the Oblique Asymptote
  1. Use Algebraic Long Division (or synthetic division) to divide \( P(x) \) by \( Q(x) \).
  2. The rational function can be written as:

    \[ R(x) = (\text{Quotient}) + \frac{\text{Remainder}}{Q(x)} \]

  3. Since \( \text{Degree}(\text{Remainder}) < \text{Degree}(Q(x)) \), as \( x \to \pm \infty \), the remainder fraction approaches zero.
  4. The O.A. is the linear quotient term: \( y = \text{Quotient} \).

Example: Find the O.A. for \( y = \frac{x^2 - 2x + 5}{x - 1} \).

Dividing \( (x^2 - 2x + 5) \) by \( (x - 1) \) yields \( (x - 1) \) with a remainder of \( 4 \).
So, \( y = (x - 1) + \frac{4}{x - 1} \).
As \( x \to \infty \), \( \frac{4}{x - 1} \to 0 \).
The Oblique Asymptote is \( y = x - 1 \).

Quick Review: Asymptotes

| Type | Condition | Method |

| Vertical | \( Q(x) = 0 \) | Set Denominator = 0 |

| Horizontal | \( \text{Deg}(P) \le \text{Deg}(Q) \) | Ratio of leading coefficients (or \( y=0 \)) |

| Oblique | \( \text{Deg}(P) = \text{Deg}(Q) + 1 \) | Algebraic division; asymptote is the quotient |

3. Essential Features for Sketching

A good sketch must show all significant features clearly and label them.

3.1 Intercepts

  • \( y \)-intercept: Set \( x = 0 \). (If \( R(0) \) exists).
  • \( x \)-intercepts (Roots): Set \( y = 0 \). This means setting the numerator \( P(x) = 0 \) and solving.

3.2 Turning Points and Stationary Points

Turning points (local maxima or minima) are where the gradient is zero. They must be shown for a detailed sketch.

Step-by-Step: Finding Turning Points
  1. Find the derivative \( \frac{dy}{dx} \) (using the quotient rule).
  2. Set \( \frac{dy}{dx} = 0 \) and solve for \( x \).
  3. Substitute these \( x \)-values back into the original function \( y=R(x) \) to find the corresponding \( y \)-coordinates.

Don't forget to check that these points do not coincide with any vertical asymptotes!

3.3 Determining the Range (Set of Values Taken by the Function)

A crucial and often tricky examination skill is determining the range of \( R(x) \). This involves using the discriminant (\( b^2 - 4ac \ge 0 \)).

Step-by-Step: Using the Discriminant

Let \( y = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials of degree at most 2.

  1. Set the function equal to a general value \( y \).
  2. Rearrange the equation to form a quadratic equation in \( x \) (in the form \( Ax^2 + Bx + C = 0 \)). Note that \( A, B, \) and \( C \) will involve the variable \( y \).
  3. For the function to exist, \( x \) must be a real value. Therefore, the discriminant must be non-negative: \( B^2 - 4AC \ge 0 \).
  4. Solve this inequality involving \( y \). The solution defines the range of the function.

Analogy: You are asking, "For which values of \( y \) can I actually find a corresponding real \( x \)?" The discriminant test checks for the existence of real solutions.

Key Takeaway:

The range calculation using the discriminant is often required when the rational function has a horizontal asymptote but still has turning points that limit the values \( y \) can take on one side of the asymptote.

4. Understanding Graph Relationships and Transformations

The syllabus requires you to understand how a basic graph \( y = f(x) \) relates to several transformed graphs. These transformations are used not only for sketching but also for solving equations and inequalities based on the original function.

4.1 Reciprocal Transformation: \( y = \frac{1}{f(x)} \)

This transformation dramatically changes the shape, focusing on the inverse relationships:

  • Where \( f(x) = 0 \) (x-intercepts of \( f(x) \)), \( y = \frac{1}{f(x)} \) has Vertical Asymptotes.
  • Where \( f(x) \) has a V.A., \( y = \frac{1}{f(x)} \to 0 \) (i.e., it approaches the x-axis).
  • Local maxima of \( f(x) \) become local minima of \( y = \frac{1}{f(x)} \) (and vice versa).
  • The sign remains the same: positive parts stay positive; negative parts stay negative.

4.2 Absolute Value Transformations

Transformation A: \( y = |f(x)| \)

The absolute value acts on the *output* (the \( y \) values):

  • Any part of the graph below the x-axis (\( y < 0 \)) is reflected symmetrically in the x-axis.
  • The positive parts (\( y \ge 0 \)) remain unchanged.
Transformation B: \( y = f(|x|) \)

The absolute value acts on the *input* (the \( x \) values):

  • The graph for negative \( x \) values (\( x < 0 \)) is removed entirely.
  • The graph for positive \( x \) values (\( x \ge 0 \)) is reflected symmetrically in the y-axis to fill the space where \( x < 0 \) was.
  • The resulting graph is always symmetric about the y-axis.

4.3 Squared Transformation: \( y^2 = f(x) \)

This graph can be written as \( y = \pm \sqrt{f(x)} \). This transformation creates symmetry about the x-axis.

  • Existence Condition: The graph only exists where \( f(x) \ge 0 \). Any part of the original graph below the x-axis is deleted.
  • Shape: The resulting graph has two components (one for \( +\sqrt{f(x)} \) and one for \( -\sqrt{f(x)} \)), symmetric about the x-axis.
  • Intercepts: The x-intercepts remain the same. The y-intercepts change from \( f(0) \) to \( \pm \sqrt{f(0)} \).

Did you know? Graphs like \( y^2 = x \) (a parabola opening sideways) are examples of this transformation applied to the linear function \( f(x)=x \).

5. Using Sketches to Solve Equations and Inequalities

Sketching graphs is often the first step in solving complex problems, especially involving inequalities.

5.1 Solving Equations Graphically

To solve an equation like \( R(x) = g(x) \) (where \( g(x) \) might be a line \( y=mx+c \) or another rational function), you sketch both \( y = R(x) \) and \( y = g(x) \) on the same axes. The solutions are the x-coordinates of the points of intersection.

5.2 Solving Inequalities Graphically

To solve \( R(x) > 0 \) or \( R(x) \le g(x) \):

  1. Identify the critical points: asymptotes, x-intercepts, and points of intersection (if two graphs are involved).
  2. Use the sketch to determine the intervals where the condition is met.
  3. Crucial Tip: When working with inequalities involving rational functions, you must always treat V.A.s as bounds for your intervals, as the function changes sign or tends to infinity across them.

Example: If you are solving \( \frac{x}{x-1} > 2 \), you sketch \( y=\frac{x}{x-1} \) and \( y=2 \). You then look for the range of \( x \) where the rational curve lies above the horizontal line.

Summary and Study Checklist

Quick Review: Sketching Rational Functions (Max Degree 2)

When approaching a rational function question, always follow this order:

  1. Simplify: Check for common factors (removable singularities/holes).
  2. Asymptotes: Find V.A. (denominator = 0) and H.A./O.A. (compare degrees, use long division if needed).
  3. Intercepts: Find where \( x=0 \) and where \( y=0 \).
  4. Turning Points/Range: Find stationary points (\( \frac{dy}{dx}=0 \)) or use the discriminant method to confirm the set of values taken by the function.
  5. Sketch: Draw the asymptotes first, plot the key points, and then draw the curve branches, ensuring the curve approaches the asymptotes correctly as \( x \to \pm \infty \) and near the V.A.s.

Practice makes perfect! The discriminant method (Section 3.3) and understanding the oblique asymptote derivation (Section 2.3) are high-yield topics for Further Mathematics Paper 1.