Welcome to Graphs of Functions!

Graphs of functions are how we visualize mathematical rules. Think of a function as a machine: you put in an input (\(x\)), and it spits out a unique output (\(y\)). Graphing is simply plotting all those inputs and outputs onto the Cartesian plane so you can see the relationship!

This chapter is vital because it connects Algebra to Geometry, helping you solve complex equations just by looking at pictures. Let's dive into the shapes and tools you'll need!

Section 1: Understanding Function Basics (C3.3 / E3.3)

1.1 Function Notation: The Math Machine Label

Instead of always writing \(y =\) something, we use function notation like \(f(x)\).

  • \(f(x)\) is pronounced "f of x" and means "the output (y-value) generated by the function \(f\) when the input is \(x\)".
  • Remember: \(y\) and \(f(x)\) mean the same thing.

Example: If the function is \(f(x) = 2x + 1\), finding \(f(3)\) means substituting \(x=3\):
\(f(3) = 2(3) + 1 = 7\). So, the point \((3, 7)\) is on the graph.

Key Takeaway: Function notation is just a professional way to say "y depends on x."

1.2 Domain and Range (Extended Content Only - E3.3)

When you look at a graph, you need to know which \(x\)-values are allowed (the Domain) and which \(y\)-values actually appear (the Range).

What is the Domain?

The Domain is the set of all possible input values (\(x\)) for which the function is defined. It describes how far left and right the graph goes.

What is the Range?

The Range is the set of all resulting output values (\(y\)). It describes how low and high the graph goes.

Analogy: Think of a photocopier. The paper you put in (x-values) is the Domain. The copies that come out (y-values) are the Range. If the photocopier only takes A4 paper, that's a restriction on the Domain!

Quick Review Box:
Domain: Horizontal movement (\(x\)).
Range: Vertical movement (\(y\)).

Section 2: Recognizing the Shapes of Graphs (C3.1 / E3.1)

Every type of function has a unique shape. Being able to recognize these shapes instantly is essential, especially when identifying functions from sketches.

2.1 Linear Functions

Form: \(f(x) = ax + b\) (or \(y = mx + c\))
Shape: A straight line.
Key Features:

  • The value \(a\) (or \(m\)) is the gradient (slope).
  • The value \(b\) (or \(c\)) is the y-intercept (where it crosses the \(y\)-axis).

2.2 Quadratic Functions

Form: \(f(x) = ax^2 + bx + c\)
Shape: A parabola (a U-shape or an inverted n-shape).
Key Features:

  • If \(a > 0\) (positive), the parabola smiles (U-shape) and has a minimum point.
  • If \(a < 0\) (negative), the parabola frowns (n-shape) and has a maximum point.
  • The turning point is called the vertex.
  • The graph is symmetrical about a vertical line passing through the vertex.

2.3 Cubic Functions (Extended Content - E3.1)

Form: \(f(x) = ax^3 + bx^2 + cx + d\)
Shape: An 'S' shape, typically starting low, curving up, then curving down, then up again (or the reverse).
Key Features:

  • Cubic graphs generally have one or two turning points (local maxima and minima).
  • They always extend from negative infinity to positive infinity along the \(y\)-axis (Range is all real numbers).

2.4 Reciprocal Functions (Extended Content - E3.1)

Form: \(f(x) = \frac{k}{x}\) (where \(k\) is a constant)
Shape: Two disconnected curves (called hyperbolas) in opposite quadrants.
Key Features:

  • The graph never touches the \(x\)-axis or the \(y\)-axis. These "invisible boundaries" are called asymptotes (E3.5).
  • The function is undefined when \(x=0\) (you cannot divide by zero!).

2.5 Exponential Functions (Extended Content - E3.1)

Form: \(f(x) = a^x\) (where \(a > 0\) and \(a \neq 1\))
Shape: A curve that grows or decays rapidly.
Key Features:

  • If \(a > 1\), the graph shows exponential growth (it rises quickly to the right).
  • If \(0 < a < 1\), the graph shows exponential decay (it falls quickly to the right).
  • It always passes through the point \((0, 1)\) because \(a^0 = 1\).
  • The \(x\)-axis is an asymptote (the graph gets close to \(y=0\) but never touches it).

2.6 Trigonometric Functions (Extended Content - E3.1)

You need to recognize, sketch, and interpret the graphs of \(y = a \sin(bx)\), \(y = a \cos(bx)\), and \(y = \tan x\) for \(0^\circ \le x \le 360^\circ\).


Sine (\(y = a \sin(bx)\)) and Cosine (\(y = a \cos(bx)\)):

  • Shape: Smooth, repeating waves (sinusoidal curves).
  • Amplitude (\(a\)): Half the distance between the maximum and minimum values. It determines the height of the wave.
  • Period (\(b\)): The length along the \(x\)-axis required for one complete cycle. For \(y = \sin x\) or \(y = \cos x\), the standard period is \(360^\circ\).
  • Differences: Sine starts at the origin \((0, 0)\). Cosine starts at its maximum point \((0, a)\).

Tangent (\(y = \tan x\)):

  • Shape: Repeats every \(180^\circ\), consisting of separate branches.
  • Asymptotes: Has vertical asymptotes at \(90^\circ\) and \(270^\circ\) (and every \(180^\circ\) thereafter).

Key Takeaway: Memorize the basic shape and two key features (like the vertex, the intercept, or the asymptote) for each function type.

Section 3: Your Graphic Display Calculator (GDC) Superpower (C3.2 / E3.2)

The GDC is essential for the 0607 syllabus. You must be fluent in using it to analyze graphs quickly.

3.1 Core GDC Skills Checklist

You must be able to use your GDC to perform the following operations:

  1. Sketch a graph: Enter the function (e.g., in Y= menu) and press Graph.
  2. Produce a table of values: Use the table feature to list coordinates for plotting points accurately.
  3. Plot points: Plotting points from a table onto a grid manually.
  4. Find zeros (x-intercepts): Find the points where \(f(x) = 0\). (Using the 'Calculate' or 'G-Solve' menu).
  5. Find local maxima or local minima: Find the turning points of curves (like the vertex of a quadratic or the hills/valleys of a cubic).
  6. Find the intersection of two graphs: Find the point \((x, y)\) where two functions, \(f(x)\) and \(g(x)\), are equal. This is often used to solve simultaneous equations graphically.
  7. Find the vertex of a quadratic: This is a specific instance of finding the local maximum or minimum (skill 5).

Don't worry if your calculator menu names are slightly different (e.g., 'G-Solve' vs 'Analyse Graph'). The key is understanding the mathematical action required. Practice finding these points regularly!

Section 4: Advanced Function Manipulation (Extended Content - E3.3)

4.1 Inverse Functions: Undoing the Process

The inverse function, written as \(f^{-1}(x)\), is the function that reverses the action of \(f(x)\).

  • If \(f(3) = 7\), then \(f^{-1}(7) = 3\).
  • Graphically, the graph of \(y = f^{-1}(x)\) is a reflection of \(y = f(x)\) in the line \(y = x\).

How to Find the Inverse Function \(f^{-1}(x)\) (Step-by-Step):

  1. Start with \(y = f(x)\). (e.g., \(f(x) = 3x - 2\), so \(y = 3x - 2\)).
  2. Swap \(x\) and \(y\) in the equation. (e.g., \(x = 3y - 2\)). (This is the critical step!)
  3. Rearrange the new equation to make \(y\) the subject. (e.g., \(x + 2 = 3y\), so \(y = \frac{x+2}{3}\)).
  4. Write your answer using inverse notation: \(f^{-1}(x) = \frac{x+2}{3}\).

4.2 Composite Functions: Function Chaining

A composite function combines two or more functions where the output of one function becomes the input of the next.

The notation \(g f(x)\) means:

  • Calculate \(f(x)\) first.
  • Use that result as the input for function \(g\).

Mnemonic Trick: Read the functions from right to left, or think of the one closest to the \(x\) being done first.

Example: If \(f(x) = x+2\) and \(g(x) = 5x\). Find \(gf(x)\).
1. Replace \(x\) in \(g(x)\) with the entire expression for \(f(x)\):
\(g(f(x)) = g(x+2)\)
2. Apply the rule of \(g\) to the new input \((x+2)\):
\(g(x+2) = 5(x+2) = 5x + 10\).

Important Note (E3.3): You are expected to give answers as a fraction in its simplest form.

Key Takeaway: Inverse functions reverse the process (\(x \leftrightarrow y\)). Composite functions chain the processes together (\(f\) then \(g\)).

Section 5: Key Features and Transformations (Extended Content - E3.4, E3.5, E3.6, E3.7)

5.1 Finding a Quadratic Function from Information (E3.4)

Sometimes you need to build the equation of a quadratic function from given points or features.

Case 1: Given Vertex \((h, k)\) and another point

Use the Vertex Form: \(y = a(x - h)^2 + k\).
Substitute \((h, k)\) and the coordinates of the other point \((x, y)\) to find the value of \(a\).

Case 2: Given x-intercepts \((\alpha, 0)\) and \((\beta, 0)\) and another point

Use the Factored Form: \(y = a(x - \alpha)(x - \beta)\).
Substitute the intercepts and the coordinates of the other point \((x, y)\) to find the value of \(a\).

5.2 Asymptotes: Invisible Boundaries (E3.5)

An asymptote is a line that a curve approaches infinitely closely, but never actually touches.

In IGCSE 0607, you primarily deal with vertical and horizontal asymptotes parallel to the axes.

  • Vertical Asymptotes (VA): These occur when the function becomes undefined, usually when the denominator of a fraction is zero. (e.g., \(f(x) = \frac{1}{x}\) has a VA at \(x=0\)).
  • Horizontal Asymptotes (HA): These describe the value the function approaches as \(x\) tends towards very large positive or negative numbers (its long-term behavior). (e.g., \(f(x) = \frac{1}{x}\) has a HA at \(y=0\)).

Did you know? The tangent graph, \(y = \tan x\), is the classic example of having many vertical asymptotes! These occur at \(x = 90^\circ, 270^\circ\), etc.

5.3 Transforming Graphs (E3.6)

If you have the graph of a function \(y = f(x)\), you can move it around the plane using simple rules. This is called translation.

Transformation 1: Vertical Translation (Up/Down)

If you have \(y = f(x) + k\):

  • The graph of \(y = f(x)\) is shifted vertically by \(k\) units.
  • If \(k\) is positive, it moves UP. If \(k\) is negative, it moves DOWN.

Example: If \(f(x) = x^2\), then \(g(x) = x^2 + 5\) shifts the parabola up by 5 units.

Transformation 2: Horizontal Translation (Left/Right)

If you have \(y = f(x + k)\):

  • The graph of \(y = f(x)\) is shifted horizontally by \(k\) units.
  • Be careful! The horizontal shift is opposite the sign of \(k\).
  • If \(k\) is positive, it moves LEFT. If \(k\) is negative, it moves RIGHT.

Example: If \(f(x) = x^2\), then \(h(x) = (x+5)^2\) shifts the parabola 5 units to the left.

5.4 The Logarithmic Function (E3.7)

The logarithmic function is the inverse of the exponential function.

  • Exponential form: \(y = a^x\)
  • Logarithmic form: \(x = \log_a y\)

At this level, we usually focus on base 10 logarithms (written as \(\log y\) on your calculator).

The relationship is:

\(10^x = y\) is equivalent to \(x = \log_{10} y\)

How it helps solve equations:

To solve an equation like \(a^x = b\), you take the log of both sides, or use the definition above. The solution is given by: \(x = \frac{\log b}{\log a}\).

Example: If \(2^x = 10\), then \(x = \frac{\log 10}{\log 2}\).

Key Takeaway: Log graphs are reflections of exponential graphs across the line \(y=x\), and they help you solve for unknown indices.


Remember: Use your GDC as a tool for checking your work and for finding precise values like intersections and vertices. Practice those calculator skills! You've got this!