Welcome to the Art of Curve Sketching!
In this chapter, we are moving away from just "plotting" points and toward "sketching" the character of a function. Think of it like this: plotting is like taking a high-resolution photograph where every pixel must be perfect, while sketching is like drawing a quick map for a friend. You don't need every tiny detail, but you do need the landmarks to be in the right place so they don't get lost!
Being able to sketch a curve is a vital skill in A Level Maths because it helps you "see" the algebra. When you look at an equation, your brain will start to translate it into a shape. Don't worry if this seems tricky at first—by the end of these notes, you'll have a step-by-step toolkit to tackle any curve the syllabus throws at you.
1. Plotting vs. Sketching: What’s the Difference?
Before we start, it is important to know exactly what the examiners are looking for when they ask you to "sketch" a curve (OCR Ref 1.02m).
- Plotting: You calculate a table of values, use graph paper, and mark points precisely.
- Sketching: You draw the general shape on plain axes. You must label key features like where the graph crosses the axes and the position of any asymptotes.
Quick Tip: Always use a pencil for the curve (in case you need to adjust the shape) and a pen for the labels!
2. Sketching Polynomials
Polynomials are functions like quadratics, cubics, and quartics. The OCR syllabus (Ref 1.02n) expects you to sketch polynomials up to degree 4 (quartics), especially when they are in factorised form.
The General Shape
The "ends" of your graph (where \(x\) gets very large or very small) are determined by the highest power of \(x\).
- Positive \(x^2\) (Quadratic): A "U" shape (Happy face).
- Positive \(x^3\) (Cubic): Starts low, ends high.
- Positive \(x^4\) (Quartic): A "W" shape.
Analogy: Think of the degree of the polynomial as its "energy level." The higher the degree, the more "wiggles" or turns the graph can have!
Finding the Landmarks (Intercepts)
To put your sketch in the right place, you need to find where it hits the axes:
- The \(y\)-intercept: Set \(x = 0\) and solve for \(y\).
- The \(x\)-intercepts (Roots): Set \(y = 0\) and solve for \(x\). This is easiest when the equation is factorised.
Repeated Roots (The "Kiss" Rule)
This is a common area where students lose marks. Pay attention to the power of the factor:
- If a factor is linear, e.g., \((x - 3)\), the graph crosses the \(x\)-axis at 3.
- If a factor is squared, e.g., \((x - 3)^2\), the graph just touches the \(x\)-axis and turns back (like a "kiss"). We call this a repeated root.
Example: Sketch \(y = (x + 1)(x - 2)^2\).
1. It’s a positive cubic (\(x \times x^2 = x^3\)), so it starts low and ends high.
2. At \(x = -1\), it crosses the axis.
3. At \(x = 2\), it touches the axis because of the squared term.
4. At \(x = 0\), \(y = (1)(-2)^2 = 4\). This is the \(y\)-intercept.
Key Takeaway:
Roots determine where the graph meets the \(x\)-axis. Single roots cross; repeated roots touch and turn.
3. Reciprocal Graphs and Asymptotes
The syllabus (OCR Ref 1.02o) requires you to know two specific reciprocal shapes: \(y = \frac{a}{x}\) and \(y = \frac{a}{x^2}\).
What is an Asymptote?
An asymptote is a straight line that the curve gets closer and closer to but never actually touches or crosses.
Analogy: Imagine you are walking toward a wall, but every step you take only covers half the remaining distance. You’ll get incredibly close to the wall, but you’ll never actually hit it!
The Shapes
- \(y = \frac{a}{x}\): Known as a hyperbola. If \(a\) is positive, the curves are in the top-right and bottom-left quadrants. It has a vertical asymptote at \(x = 0\) and a horizontal one at \(y = 0\).
- \(y = \frac{a}{x^2}\): This looks like a "volcano" or a "chimney." Since \(x^2\) is always positive (for \(x \neq 0\)), the graph stays above the \(x\)-axis in both the top-left and top-right quadrants.
Did you know? We can't have \(x = 0\) in these equations because "thou shalt not divide by zero!" That's exactly why there is a vertical "gap" (asymptote) at \(x = 0\).
4. The Modulus Function: \(y = |ax + b|\)
The modulus (or absolute value) is like a "positivity machine." Whatever number you put in, it spits out the positive version. The notation is \(|x|\) (OCR Ref 1.02s).
How to Sketch \(y = |f(x)|\)
- Sketch the "normal" line \(y = ax + b\) using a light dotted line.
- Any part of the line that is below the \(x\)-axis (where \(y\) is negative) must be reflected up to become positive.
- The result is usually a V-shape.
Quick Review: The vertex (the pointy bit) of the V-shape occurs where the stuff inside the modulus bars equals zero. For \(y = |x - 3|\), the point is at \(x = 3\).
5. Intersection Points and Solving Equations
You can use your sketching skills to solve equations or inequalities (OCR Ref 1.02q, 1.02t). If you are asked to solve \(f(x) = g(x)\), you are looking for the intersection points—the places where the two sketches cross each other.
Common Mistake to Avoid:
When solving modulus equations like \(|x + 2| = 5\), many students only find one answer. By sketching \(y = |x + 2|\) and the horizontal line \(y = 5\), you can clearly see there are two intersection points! Always check your sketch to see how many solutions you should be looking for.
6. Proportional Relationships
The syllabus (Ref 1.02r) mentions relating graphs to variation.
- Direct Proportion (\(y \propto x\)): This is a straight line through the origin, \(y = kx\).
- Inverse Proportion (\(y \propto \frac{1}{x}\)): This is the reciprocal graph shape we studied in Section 3.
Summary Checklist for Curve Sketching
Before you finish a sketch, ask yourself these four questions:
- Shape: Does it have the right general shape for the degree (e.g., cubic, reciprocal)?
- Intercepts: Have I clearly labeled the \(y\)-intercept and all \(x\)-intercepts?
- Asymptotes: If the graph has asymptotes, have I drawn them as dashed lines and written their equations (e.g., \(x = 0\))?
- Behavior: Does the graph cross the axis or "kiss" it at the roots?
Key Takeaway: A sketch is a visual summary of algebraic properties. Accuracy in labeling "key points" is more important than artistic talent!