Mathematics (0580) | Sketching curves

Mastering the Art of Sketching Curves (Algebra and Graphs)

Hello future mathematicians! Sketching curves is one of the most powerful skills in the "Algebra and Graphs" section. Why? Because a sketch gives you an instant, visual understanding of an algebraic equation.
It helps you see the solutions, turning points, and long-term behaviour of a function, even before you calculate them exactly. Don't worry if this seems tricky at first; we will break down the fundamental shapes and the essential features you need to capture to draw a perfect mathematical snapshot!

This topic heavily relies on using your **Graphic Display Calculator (GDC)**, so make sure you are comfortable with its basic graphing functions (covered in Section 4).

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Section 1: The Essential Gallery of Function Shapes

A "sketch" is not the same as a detailed, scaled plot. A good sketch shows the correct shape and highlights the important features like intercepts and turning points.

1.1 Linear Functions (The Straight Line)

This is the simplest function.

The equation is given by: \(f(x) = ax + b\) or \(y = mx + c\)

• Shape: A straight line.
• Gradient (\(m\) or \(a\)): Determines the steepness. If \(m > 0\), the line slopes up (positive correlation). If \(m < 0\), it slopes down (negative correlation).
• Y-intercept (\(c\) or \(b\)): Where the line crosses the \(y\)-axis (where \(x=0\)).

Quick Tip: To sketch a linear graph, you only need two points! The intercepts (where it crosses the x and y axes) are usually the easiest to find.

1.2 Quadratic Functions (The Parabola)

The quadratic function produces the classic U-shape or inverted U-shape.

The general form is: \(f(x) = ax^2 + bx + c\)

• Shape: A parabola.
• Direction:
  • If \(a > 0\) (positive), the parabola opens upwards (it's a 'happy face', minimum turning point).
  • If \(a < 0\) (negative), the parabola opens downwards (it's a 'sad face', maximum turning point).
• Vertex: The turning point (maximum or minimum).

1.3 Cubic Functions (The S-Shape)

Cubic functions have up to two turning points and typically look like a wavy line or an 'S' that has been stretched.

The general form is: \(f(x) = ax^3 + bx^2 + cx + d\)

• Shape: An extended 'S' shape, often called a cubic curve.
• Turning Points: Can have zero, one (a point of inflection), or two turning points (a local maximum and a local minimum).
• End Behavior: If \(a\) is positive, the graph goes from bottom-left to top-right. If \(a\) is negative, it goes from top-left to bottom-right.

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Section 2: Finding Essential Features for Sketching

When sketching any function, you must clearly mark these three features:

2.1 Intercepts (Where the Graph Crosses the Axes)

Intercepts are easy to calculate and vital for setting up your sketch axes correctly.

a) Y-intercept (Where \(x=0\))

To find where the graph crosses the \(y\)-axis, substitute \(x=0\) into the equation.
Example: For \(y = x^2 - 3x + 2\):
Set \(x=0 \rightarrow y = (0)^2 - 3(0) + 2 = 2\). The y-intercept is \((0, 2)\).

b) X-intercepts or Zeros (Where \(y=0\))

To find where the graph crosses the \(x\)-axis, set \(y=0\) and solve the resulting equation. These points are also called the **roots** or **zeros** of the function.
Example: For \(0 = x^2 - 3x + 2\):
Factorise: \(0 = (x-1)(x-2)\). The x-intercepts are \((1, 0)\) and \((2, 0)\).

2.2 Turning Points (Local Maxima and Minima)

The **turning point** (or **vertex** for a quadratic) is where the graph changes direction.

Finding the Vertex of a Quadratic Algebraically

For the quadratic \(y = ax^2 + bx + c\), the coordinates of the vertex are:

1. Find the x-coordinate using the axis of symmetry formula:
   \[x = -\frac{b}{2a}\]
2. Substitute this \(x\)-value back into the original equation to find the corresponding \(y\)-coordinate.

Example: Find the vertex of \(y = x^2 - 4x + 1\).
1. \(a=1, b=-4\). \(x = -\frac{(-4)}{2(1)} = 2\).
2. Substitute \(x=2\): \(y = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3\).
The vertex is \((2, -3)\). Since \(a=1\) (positive), this is a minimum point.

Access for Struggling Students: If the algebraic method seems confusing, remember that your Graphic Display Calculator (GDC) can find the vertex (Local Maxima or Local Minima) for you easily (C3.2(d), C3.2(f)).

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Section 3: Using the Graphic Display Calculator (GDC) for Sketching

The GDC is an essential tool for IGCSE Mathematics (0607). You must be able to use it effectively to analyse functions before you sketch them by hand.

GDC Skills Required (C3.2 / E3.2)

You should be able to use your GDC to:

• (a) Sketch the graph: Quickly display the function shape.
• (b) Produce a table of values: Useful if the function is unfamiliar, helping you plot a few key points.
• (d) Find Zeros: Calculate the exact coordinates of the x-intercepts (\(y=0\)).
• (d) Find Local Maxima or Local Minima: Determine the coordinates of the turning points (vertices).
• (e) Find the Intersection of two graphs: Find where two functions \(y = f(x)\) and \(y = g(x)\) cross. This is equivalent to solving \(f(x) = g(x)\) graphically.
• (f) Find the vertex of a quadratic: A specific calculator function dedicated to the turning point of a parabola.

Did you know? Finding the intersection of two graphs on your GDC is often the quickest way to solve complex or unfamiliar equations like \(2x = x^2\) (C2.5.4) or \(2x - 1 = 1/x\) (E2.5.7).

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Section 4: Advanced Curves and Features (Extended Content E3.1, E3.5, E3.6)

Extended students need to recognise a wider range of function shapes and understand key concepts like asymptotes and transformations.

4.1 Extended Function Shapes

In addition to linear, quadratic, and cubic graphs, Extended students must recognise:

• Reciprocal: \(f(x) = \frac{k}{x}\) (Hyperbola).
• Exponential: \(f(x) = a^x\) (Growth or Decay curve).
• Trigonometric: \(f(x) = a \sin(bx)\), \(a \cos(bx)\), \(\tan x\).

4.2 Understanding Asymptotes (E3.5)

An **asymptote** is a straight line that a curve approaches infinitely closely but never touches or crosses.

• Definition Analogy: Think of two parallel train tracks. The train gets closer and closer to the track beside it, but they never merge. The tracks represent the graph and the asymptote.
• Vertical Asymptotes: These occur when the function is undefined, typically because the denominator of a fraction is zero.
  Example: For \(f(x) = \frac{1}{x}\), if \(x=0\), the function is undefined. Thus, \(x=0\) (the \(y\)-axis) is a vertical asymptote.
• Horizontal Asymptotes: These describe the **long-term behaviour** of the graph (what \(y\) approaches as \(x\) gets very large or very small).
  Example: For \(f(x) = \frac{1}{x}\), as \(x\) gets huge, \(1/x\) gets closer to zero. Thus, \(y=0\) (the \(x\)-axis) is a horizontal asymptote.

The syllabus specifically mentions asymptotes for the trigonometric function \(f(x) = \tan x\), which has vertical asymptotes at \(90^\circ\), \(270^\circ\), etc.

4.3 Transforming Graphs (Translations) (E3.6)

Transformations describe how a graph is moved from a basic function \(y = f(x)\). For IGCSE, you only need to know about simple **translations** (shifts).

Type 1: Vertical Shift (Moving Up or Down)

Equation: \(y = f(x) + k\)

• The entire graph of \(y = f(x)\) is shifted vertically.
• If \(k\) is positive, the graph shifts **up** by \(k\) units.
• If \(k\) is negative, the graph shifts **down** by \(k\) units.

Example: If \(f(x) = x^2\), then \(y = x^2 + 3\) is the parabola shifted 3 units up.

Type 2: Horizontal Shift (Moving Left or Right)

Equation: \(y = f(x + k)\)

• The entire graph of \(y = f(x)\) is shifted horizontally.
• If \(k\) is positive, the graph shifts **left** by \(k\) units (this is counter-intuitive, think of \(x+k=0 \Rightarrow x=-k\)).
• If \(k\) is negative, the graph shifts **right** by \(k\) units.

Example: If \(f(x) = x^2\), then \(y = (x - 2)^2\) is the parabola shifted 2 units to the right.

Memory Aid: "Vertical is True, Horizontal is a Lie."
(Vertical moves exactly as the sign suggests, horizontal moves the opposite way of the sign).

4.4 Finding the Quadratic Function (Extended E3.4)

If you are given the key features of a quadratic graph, you must be able to construct its equation.

a) Given the Vertex \((h, k)\) and another point

Use the **Vertex Form**:
\[y = a(x - h)^2 + k\]
1. Substitute the vertex coordinates \((h, k)\) into the formula.
2. Substitute the coordinates of the other given point \((x, y)\) to find the scaling factor \(a\).
3. Write the final equation.

b) Given the X-intercepts (\(p, 0\)) and (\(q, 0\)) and another point

Use the **Intercept (Root) Form**:
\[y = a(x - p)(x - q)\]
1. Substitute the intercepts \(p\) and \(q\) into the formula.
2. Substitute the coordinates of the other given point \((x, y)\) to find the scaling factor \(a\).
3. Write the final equation.