Further Pure Mathematics | Graphs

Welcome to the Graphs Chapter!

Hello future Further Pure Mathematician! This chapter on Graphs is one of the most visual and rewarding topics you will study. Think of it as learning the language of mathematical pictures.
In this chapter, we will master how to sketch complicated curves quickly, understand how small changes to an equation dramatically transform its graph, and deal with functions involving absolute values.

Don't worry if sketching seems difficult at first. We will break down every technique into simple, manageable steps. Mastering these skills is vital because graphing helps us solve equations, understand calculus concepts, and visualize mathematical problems. Let’s dive in!

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Section 1: Transformations of Graphs

When you start with a known graph, like \(y = x^2\) or \(y = \sin x\), transformations tell you exactly how the graph moves, stretches, or flips when you adjust the equation.

The Golden Rule: Inside vs. Outside

This is the key to understanding transformations. Think of the function \(y = f(x)\):

• Outside Changes (Affecting \(y\)): If you add, subtract, or multiply outside the function notation (e.g., \(y = f(x) + a\)), the effect is intuitive (it does exactly what you expect) and affects the vertical direction (y-axis).
• Inside Changes (Affecting \(x\)): If you add, subtract, or multiply inside the function notation (e.g., \(y = f(x + a)\)), the effect is counter-intuitive (it does the opposite of what you expect) and affects the horizontal direction (x-axis).

1. Translations (Shifts)

Translation simply means sliding the graph without changing its shape or size.

• Vertical Translation: \(y = f(x) + a\)

  The graph moves \(a\) units vertically. If \(a\) is positive, it moves up. If \(a\) is negative, it moves down. (Intuitive, Outside)

  Example: \(y = x^2 + 3\) moves the graph of \(y = x^2\) 3 units up.

• Horizontal Translation: \(y = f(x + a)\)

  The graph moves \(a\) units horizontally. This is the tricky one! If \(a\) is positive (e.g., \(x+3\)), the graph moves left (negative direction). If \(a\) is negative (e.g., \(x-3\)), the graph moves right (positive direction). (Counter-intuitive, Inside)

  Memory Aid: Think of the horizontal shift as "Opposite Day." If you see \(+3\), you go left \(-3\). If you see \(-5\), you go right \(+5\).

2. Stretches and Compressions (Scaling)

A stretch changes the shape of the graph, making it narrower or wider/taller or flatter.

• Vertical Stretch: \(y = a f(x)\)

  The graph is stretched away from the x-axis by a scale factor of \(a\). If \(a > 1\), it gets taller. If \(0 < a < 1\), it gets flatter (compression). (Intuitive, Outside)

  Every y-coordinate is multiplied by \(a\).

• Horizontal Stretch: \(y = f(ax)\)

  The graph is stretched away from the y-axis by a scale factor of \(\frac{1}{a}\). This is another tricky one! If \(a=2\), the graph is compressed by a factor of \(\frac{1}{2}\). (Counter-intuitive, Inside)

  Every x-coordinate is divided by \(a\).

3. Reflections

Reflections are a special type of stretch where the scale factor is \(-1\).

• Reflection in the x-axis: \(y = -f(x)\)

  The graph is flipped vertically across the x-axis. (Outside)

  Analogy: The x-axis acts like a mirror placed horizontally.

• Reflection in the y-axis: \(y = f(-x)\)

  The graph is flipped horizontally across the y-axis. (Inside)

  Analogy: The y-axis acts like a mirror placed vertically.

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Section 2: The Modulus Function (Absolute Value)

The modulus function, denoted by \(|x|\), gives the absolute value of \(x\). In simple terms, it always returns the non-negative distance of the number from zero.

Definition of the Modulus

Key Term: The modulus of a number is its value without regard to its sign.

$$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

Example: \(|5| = 5\) and \(|-5| = 5\).

Sketching Graphs involving Modulus

The modulus sign can be applied in two main places, and the sketching technique is completely different for each.

1. Sketching \(y = |f(x)|\)

This transformation affects the output (y-value) of the function. Since the resulting y-value must always be non-negative, any part of the graph that dips below the x-axis must be flipped up.

Step-by-Step Process for \(y = |f(x)|\):

1. First, sketch the original graph \(y = f(x)\) completely.
2. Identify any parts of the graph that are below the x-axis (where \(y < 0\)).
3. Reflect these negative parts of the graph in the x-axis (flip them upwards).
4. The part of the graph that was already above or on the x-axis remains unchanged.

Analogy: Imagine the x-axis is a floor. If any part of the graph falls through the floor, it bounces straight back up, symmetrically!

2. Sketching \(y = f(|x|)\)

This transformation affects the input (x-value). Since \(|x|\) means that we get the same output whether the input is \(x\) or \(-x\) (e.g., \(f(|3|)\) is the same as \(f(|-3|)\)), the graph must be symmetrical about the y-axis.

Step-by-Step Process for \(y = f(|x|)\):

1. First, sketch the original graph \(y = f(x)\) completely.
2. Delete the entire part of the graph that exists where \(x\) is negative (the whole left side, where \(x < 0\)).
3. Keep the part of the graph that exists where \(x \ge 0\) (the right side).
4. Reflect the remaining right-hand side (the part where \(x \ge 0\)) in the y-axis to create the new left-hand side.

Common Mistake to Avoid: Do NOT mix up the two types! \(|f(x)|\) flips negative y-values (affects the bottom half), while \(f(|x|)\) deletes the negative x-side and mirrors the positive x-side (affects the left half).

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Section 3: Sketching Polynomials

Further Pure Maths often requires you to sketch cubic (\(x^3\)) and quartic (\(x^4\)) functions based on their roots (where they cross the x-axis).

1. General Shapes

The overall shape of a polynomial is determined by two things: the degree (highest power of \(x\)) and the sign of the leading coefficient (the number in front of the highest power).

a) Cubic Functions (\(y = ax^3 + \dots\))

• Positive leading coefficient (\(a > 0\)): The graph starts low (bottom-left) and ends high (top-right). It looks like a chair going up hill.
• Negative leading coefficient (\(a < 0\)): The graph starts high (top-left) and ends low (bottom-right). It looks like a slide or a chair going down hill.

b) Quartic Functions (\(y = ax^4 + \dots\))

Quartics are symmetrical in their end behavior – both ends point in the same direction.

• Positive leading coefficient (\(a > 0\)): The graph starts high and ends high. It generally has a 'W' shape (or sometimes a 'U' shape).
• Negative leading coefficient (\(a < 0\)): The graph starts low and ends low. It generally has an 'M' shape (or sometimes an upside-down 'U' shape).

2. Dealing with Roots (Intercepts)

To accurately sketch the shape, you must identify where the graph crosses the x-axis. These points are called the roots of the equation \(f(x) = 0\).

The Multiplicity of Roots

When a factor is repeated in the equation (e.g., \((x-r)^n\)), this affects how the graph behaves at the root \(x=r\):

• Single Root (Odd Multiplicity, e.g., \((x-r)^1\), \((x-r)^3\)):

  The graph crosses the x-axis cleanly at \(x=r\).

• Double Root (Even Multiplicity, e.g., \((x-r)^2\), \((x-r)^4\)):

  The graph touches the x-axis at \(x=r\) and immediately turns around (it is a turning point on the x-axis).

Step-by-Step Sketching Polynomials:

1. Find the y-intercept by setting \(x=0\).
2. Find the x-intercepts (roots) by setting \(y=0\) (often requires factorizing).
3. Determine the end behavior (the general shape) based on the degree and leading coefficient.
4. Use the multiplicity of the roots to determine whether the graph crosses or touches at each x-intercept.
5. Sketch the curve, ensuring it flows correctly from its start point, through the intercepts, and towards its end point.

Example: Sketch \(y = (x-2)^2 (x+1)\).
Degree is 3 (cubic). Leading coefficient is positive.
Roots: \(x=2\) (double root, touches) and \(x=-1\) (single root, crosses).
Y-intercept: \(y = (-2)^2(1) = 4\).
Sketch starts low, crosses at \(-1\), goes up to cross the y-axis at 4, hits 2, touches, and turns up high.

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Section 4: Sketching Rational Functions (Asymptotes)

A rational function is a function that can be written as the ratio (a fraction) of two polynomials, \(y = \frac{P(x)}{Q(x)}\). These graphs often contain features called asymptotes, which are lines that the graph approaches but never touches.

1. Vertical Asymptotes (V.A.)

A vertical asymptote occurs when the denominator \(Q(x)\) is zero, but the numerator \(P(x)\) is not zero. At this point, the function value approaches infinity (it is undefined).

How to find V.A.: Set the denominator equal to zero and solve for \(x\).
Example: For \(y = \frac{x+1}{x-3}\), the V.A. is \(x=3\).

2. Horizontal Asymptotes (H.A.)

A horizontal asymptote describes the behavior of the graph as \(x\) becomes extremely large (positive or negative, i.e., \(x \to \pm \infty\)). We look at the degree (highest power) of the numerator and the denominator. Let \(n\) be the degree of \(P(x)\) and \(m\) be the degree of \(Q(x)\).

• Case 1: Degree of Numerator < Degree of Denominator (\(n < m\))

  The denominator grows much faster than the numerator. The H.A. is always \(y = 0\) (the x-axis).

  Example: \(y = \frac{x^2}{x^3 - 1}\). (2 < 3, so H.A. is \(y=0\)).

• Case 2: Degree of Numerator = Degree of Denominator (\(n = m\))

  The H.A. is the line \(y = \frac{\text{Leading Coefficient of Numerator}}{\text{Leading Coefficient of Denominator}}\).

  Example: \(y = \frac{4x^2 - 5}{2x^2 + x}\). (2 = 2, so H.A. is \(y = \frac{4}{2} = 2\)).

• Case 3: Degree of Numerator > Degree of Denominator (\(n > m\))

  There is no horizontal asymptote (the graph tends towards infinity or negative infinity). In some curricula, this leads to an oblique/slant asymptote, but for standard IGCSE FP, we just state there is no H.A. The graph continues growing without bound.

3. Sketching Rational Functions Summary

To sketch \(y = \frac{P(x)}{Q(x)}\), follow these steps:

1. Find Vertical Asymptotes (Denom = 0).
2. Find Horizontal Asymptotes (Compare degrees).
3. Find x-intercepts (Num = 0).
4. Find y-intercept (Set \(x=0\)).
5. Use the intercepts and asymptotes to determine the shape in each region defined by the V.A.

Did you know? Rational functions often look like the basic hyperbola \(y = \frac{1}{x}\), which has asymptotes \(x=0\) and \(y=0\).

Dealing with Intersections of Asymptotes

The point where the H.A. and V.A. intersect is the "centre" of the graph. The graph is often sketched in relation to these two guiding lines. For example, the graph of \(y = \frac{1}{x-2} + 3\) has asymptotes \(x=2\) and \(y=3\), meaning the entire graph of \(y = \frac{1}{x}\) has been translated by the vector \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\).

Congratulations! You now have a solid foundation in the essential graphing techniques required for Further Pure Mathematics. Keep practicing those modulus flips and asymptote rules!