Graphing Polynomial and Reciprocal Functions

Hello! Welcome to the first lesson in our module on Graphs of Functions and Transformations.

In our last lesson, we concluded the Coordinate Geometry module by mastering the multi-step process for finding the equation of a perpendicular bisector. That required combining your skills in finding midpoints, gradients, and forming linear equations. Today, we'll build on that foundation of graphical thinking as we move beyond straight lines.

This lesson addresses a topic you've identified as challenging: graphs. Our learning outcome is to sketch and identify key features of quadratic, cubic, and reciprocal functions. We will focus on developing a systematic, exam-focused approach to sketching, which is different from simply plotting points. A sketch captures the essential properties of a graph—its overall shape, where it crosses the axes, and its turning points—without needing to be perfectly to scale.

What are the "Key Features" of a Graph?

When an exam asks you to sketch a graph, it's testing your ability to quickly identify and label its most important characteristics. These are the "key features" that define the graph's behaviour.

Key Features of a Graph Sketch — This image shows a typical cubic graph. The key features you are expected to identify are clearly labelled: the y-intercept (where it crosses the y-axis), the roots (where it crosses the x-axis), and any maximum or minimum turning points.

Our goal is to learn a checklist of these features for each function type, so you can produce accurate sketches quickly and efficiently in the exam.

1. Quadratic Functions: Parabolas

You'll recognize quadratic functions from the x² term. Their graphs are smooth, symmetrical U-shaped curves called parabolas.

Plotting vs. Sketching

A good way to build intuition for the shape of these graphs is by plotting them from a table of values. However, for exams, sketching is a much faster and more powerful skill.

Let's start by watching how to plot a quadratic. This will help solidify the fundamental shape in your mind and highlight the importance of careful calculation, especially with negative numbers—an area you've noted for improvement.

Drawing Quadratic, Cubic & Reciprocal Graphs | Grade 5+ Series | GCSE Maths Tutor

This video from 'The GCSE Maths Tutor' demonstrates the plotting method. It's a great refresher on how the equation generates the graph's shape.

Watch the section on quadratics from 00:01:24 to 00:07:40 and from 00:12:30 to 00:14:44. Pay attention to: How to substitute negative x-values correctly (e.g., (-3)² not -3²). How a positive x² term creates a U-shape (a 'minimum' point). How a negative x² term (4 - x²) flips the graph into an n-shape (a 'maximum' point).

A Checklist for Sketching Quadratics

Plotting is too slow for an exam. A sketch requires a more analytical approach. The "Ginger Mathematician" explains a systematic method for finding the key features needed for a sketch. This also introduces a key strategy for choosing the most efficient method to find the turning point.

All of iGCSE Maths Sketching Graphs: What You Need To Know

This video focuses on the exam skill of sketching. It presents a clear checklist and demonstrates two different methods for finding the vertex, which is excellent practice for method selection.

Watch the detailed section on sketching quadratics from 00:07:56 to 00:26:13. Focus on this checklist: Shape: Determined by the sign of the x² term (a). a > 0 gives a 'happy face' (U), a < 0 gives a 'sad face' (∩). Y-intercept: Find this by setting x = 0. Roots (X-intercepts): Find these by setting y = 0 and solving the quadratic equation. Vertex (Turning Point): Note the two methods discussed for finding the x-coordinate of the vertex: Using the axis of symmetry formula: x = -b / (2a) Completing the square.

The formula $x = - b / (2 a)$ is a very powerful and efficient tool for finding the vertex. It's not on the formula sheet but is derived from the quadratic formula and is worth memorizing to save time.

Let's consolidate these ideas.

Cambridge Syllabus Notes - Quadratic Functions

These notes from e-Consult Kenya provide a concise summary of the key properties of quadratic functions, including the different forms of the equation and the formula for the axis of symmetry.

Read section '2. Quadratic Functions'. This will reinforce the concepts of standard form, vertex form, the axis of symmetry formula, and intercepts.

Test your understanding!

Sketch the graph of $y = x^{2} - 4 x - 5$ . Label the coordinates of the vertex and any points where the graph intersects the axes.

Checklist:

Determine the shape.
Find the y-intercept.
Find the x-intercepts (roots).
Find the coordinates of the vertex.

Show answer

Shape: The coefficient of $x^{2}$ is +1, so the graph is a U-shaped parabola.
Y-intercept: Set $x = 0$ . $y = 0^{2} - 4 (0) - 5 = - 5$ . The y-intercept is at $(0, - 5)$ .
X-intercepts (Roots): Set $y = 0$ . $x^{2} - 4 x - 5 = 0$ . This factorizes easily: $(x - 5) (x + 1) = 0$ . So, the roots are at $x = 5$ and $x = - 1$ . The intercepts are at $(5, 0)$ and $(- 1, 0)$ .
Vertex: Let's use the efficient formula $x = - b / (2 a)$ . Here, $a = 1$ and $b = - 4$ .
$x_{v er t e x} = \frac{- ( - 4 )}{2 ( 1 )} = \frac{4}{2} = 2$
To find the y-coordinate, substitute $x = 2$ back into the equation:
$y_{v er t e x} = (2)^{2} - 4 (2) - 5 = 4 - 8 - 5 = - 9$
The vertex is at $(2, - 9)$ .

Your sketch should be a U-shaped curve passing through $(- 1, 0)$ , $(5, 0)$ , and $(0, - 5)$ , with its minimum point at $(2, - 9)$ .

2. Cubic Functions

Cubic functions, which have an $x^{3}$ term, typically have a characteristic 'S' shape. For IGCSE, you'll most often be asked to sketch them when they are given in a convenient factorized form.

Sketching from Factors

The factorized form makes finding the key features very straightforward.

All of iGCSE Maths Sketching Graphs: What You Need To Know

Let's return to the 'Ginger Mathematician' video, which covers exactly the type of cubic sketching question you'll see in an exam.

Watch the section on cubic functions from 00:27:56 to 00:38:14, and then the specimen paper example from 00:39:46 to 00:43:49. Focus on this checklist: Shape: Determined by the sign of the x³ term. +x³ gives an 'uphill' shape (starts low, ends high), while -x³ gives a 'downhill' shape. Roots (X-intercepts): These can be read directly from the factors. Y-intercept: Find this by setting x = 0.

As you saw, if the equation is something like $y = x (x - 2) (x + 3)$ , the roots are immediately obvious: $x = 0$ , $x = 2$ , and $x = - 3$ . This is a prime example of where recognizing the form of the equation saves you a huge amount of time.

3. Reciprocal Functions

Reciprocal functions have the form $y = k / x$ . Their graphs are called hyperbolas and have a unique shape consisting of two separate branches.

Asymptotes

The most important feature of a reciprocal graph is its asymptotes. These are lines that the graph gets closer and closer to but never actually touches.

For the basic graph $y = k / x$ :

The graph never touches the y-axis because you cannot divide by zero ( $x \neq = 0$ ). The y-axis is a vertical asymptote.
The graph never touches the x-axis because $k / x$ can never equal zero (unless $k = 0$ , which is trivial). The x-axis is a horizontal asymptote.

Drawing Quadratic, Cubic & Reciprocal Graphs | Grade 5+ Series | GCSE Maths Tutor

This clip from 'The GCSE Maths Tutor' provides a clear, practical explanation of the reciprocal graph's shape and its relationship with the axes.

Watch from 00:22:19 to 00:26:15. Note how the two branches are formed and the explanation of why the graph never touches the axes (the concept of asymptotes).

The location of the branches depends on the sign of $k$ :

If $k > 0$ (e.g., $y = 4/ x$ ), the branches are in the first and third quadrants.
If $k < 0$ (e.g., $y = - 4/ x$ ), the branches are in the second and fourth quadrants.

Conclusion

Today, you've learned to move beyond plotting points and have started developing the analytical skills needed to sketch key functions efficiently. This is vital for saving time and securing marks in your exams.

Key Takeaways:

Sketching is about capturing the key features: shape, intercepts, and turning points.
Quadratic graphs ( $a x^{2} + ...$ ) are parabolas. Their shape (U or ∩) depends on the sign of a. Key features are roots, y-intercept, and the vertex (which you can find efficiently using $x = - b /2 a$ ).
Cubic graphs ( $a x^{3} + ...$ ) have an 'S' shape. Their direction ('uphill' or 'downhill') depends on the sign of a. In exams, they are often factorized, making the roots easy to find.
Reciprocal graphs ( $k / x$ ) are hyperbolas with two branches. Their key features are the asymptotes (the x and y axes for the basic form) and the quadrants their branches lie in (determined by the sign of k).

For practice, which is central to your "Error-Loop Method", work through some past paper questions on this topic.

IGCSE Questions by Topic

The 'Past Paper Penguin' site has excellent collections of exam questions by topic. This is your chance to apply the checklist methods we've covered today.

Download the PDFs for 'Sketching Graphs of Functions' and 'Cubics'. Attempt several questions from each. For every question, consciously use the checklists we've discussed. Then, use the provided solutions to check your work and identify any gaps in your logic.

Preview of the Next Lesson:
Now that you can sketch these fundamental graphs, the next logical step is to use them. In our next lesson, we will learn how to solve equations graphically by finding the coordinates of intersection points between two graphs.

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