This article covers the essential skills of quadratic functions and their graphs – a core topic in DSE Paper 1 Section A(2) and Section B worth 4–6 marks. You will learn how to find the vertex and axis of symmetry of a parabola, how to sketch quadratic graphs, how to complete the square to convert between general and vertex forms, and how to apply graph transformations to quadratic functions. The article includes step-by-step worked examples, DSE exam techniques, practice questions. These are essential skills that appear regularly in DSE papers.
Learning Objectives
By the end of this article, you should be able to:
- Identify the vertex and axis of symmetry of a quadratic function.
- Convert a quadratic function from general form \( ax^2 + bx + c \) to vertex form \( a(x-h)^2 + k \) by completing the square.
- Sketch the graph of a quadratic function, identifying the vertex, axis of symmetry, and intercepts.
- Apply graph transformations (translations, stretches, reflections) to quadratic functions.
- Determine the nature of the roots (discriminant) from the graph.
- Solve DSE-style problems involving quadratic functions and their graphs.
1. Introduction:
Quadratic functions and their graphs appear every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:
- Find the vertex or axis of symmetry of a parabola
- Sketch the graph of a quadratic function
- Complete the square to find the maximum or minimum value
- Apply transformations to quadratic graphs
These are essential skills that also appear in more advanced topics.
DSE Exam Tip
Quadratic function questions often appear in Section A(2) as short-answer questions worth 3–4 marks, and in Section B as longer problems worth 5–6 marks.
2. Forms of Quadratic Functions
A quadratic function can be written in two main forms:
- General form: \( f(x) = ax^2 + bx + c \) (where \( a \neq 0 \))
- Vertex form: \( f(x) = a(x - h)^2 + k \) – the vertex is \( (h, k) \)
The vertex form is very useful because it directly gives the vertex and the axis of symmetry.
Vertex and Axis of Symmetry
- Vertex: \( (h, k) \)
- Axis of symmetry: \( x = h \)
Example
For \( f(x) = 2(x - 3)^2 + 5 \):
Vertex: \( (3, 5) \)
Axis of symmetry: \( x = 3 \)
3. Completing the Square
To convert from general form to vertex form, we use completing the square.
Steps
- Factor out \( a \) from the \( x^2 \) and \( x \) terms.
- Inside the bracket, add and subtract \( \left(\frac{b}{2a}\right)^2 \).
- Rewrite the perfect square trinomial.
- Simplify to vertex form.
Worked Example
Convert: \( f(x) = 2x^2 - 8x + 3 \) to vertex form.
Solution
\( f(x) = 2(x^2 - 4x) + 3 \)
Complete the square inside: \( x^2 - 4x = (x - 2)^2 - 4 \)
\( f(x) = 2[(x - 2)^2 - 4] + 3 \)
\( f(x) = 2(x - 2)^2 - 8 + 3 \)
\( f(x) = 2(x - 2)^2 - 5 \)
Vertex: \( (2, -5) \)
4. Sketching Quadratic Graphs
To sketch a quadratic graph, you need to find:
- Vertex – the turning point
- Axis of symmetry – the vertical line through the vertex
- y-intercept – \( f(0) = c \)
- x-intercepts (if any) – solve \( f(x) = 0 \)
- Direction of opening – if \( a > 0 \), opens upward; if \( a < 0 \), opens downward.
Example
Sketch \( f(x) = (x-1)^2 - 4 \).
Vertex: \( (1, -4) \)
Axis: \( x = 1 \)
y-intercept: \( f(0) = (0-1)^2 - 4 = 1 - 4 = -3 \)
x-intercepts: \( (x-1)^2 - 4 = 0 \) → \( (x-1)^2 = 4 \) → \( x-1 = \pm 2 \) → \( x = 3 \) or \( x = -1 \)
Opens upward (\( a = 1 > 0 \)).
5. Graph Transformations of Quadratics
Transformations of the basic parabola \( y = x^2 \) can be described using vertex form:
- \( y = a(x - h)^2 + k \) is the graph of \( y = x^2 \) after:
- Horizontal translation by \( h \) units (right if \( h > 0 \), left if \( h < 0 \))
- Vertical translation by \( k \) units (up if \( k > 0 \), down if \( k < 0 \))
- Vertical stretch (if \( |a| > 1 \)) or compression (if \( 0 < |a| < 1 \))
- Reflection across the x-axis if \( a < 0 \)
DSE Memory Aid
Vertex form: \( a(x-h)^2 + k \) – \( (h,k) \) is the vertex. Remember: \( h \) is the horizontal shift, \( k \) is the vertical shift.
6. Maximum and Minimum Values
The vertex gives the maximum or minimum value of the quadratic function:
- If \( a > 0 \), the parabola opens upward → the vertex is a minimum point.
- If \( a < 0 \), the parabola opens downward → the vertex is a maximum point.
The maximum or minimum value is the \( y \)-coordinate of the vertex.
Example
For \( f(x) = -2(x-1)^2 + 3 \):
\( a = -2 < 0 \) → opens downward → maximum at vertex \( (1, 3) \)
Maximum value: \( 3 \)
7. Worked Examples
Example 1: Finding Vertex and Axis
Question: Find the vertex and axis of symmetry of \( f(x) = -3x^2 + 6x - 2 \).
Solution
\( a = -3, b = 6, c = -2 \)
\( h = -\frac{b}{2a} = -\frac{6}{2(-3)} = 1 \)
\( k = f(1) = -3(1)^2 + 6(1) - 2 = -3 + 6 - 2 = 1 \)
Vertex: \( (1, 1) \)
Axis: \( x = 1 \)
Example 2: Completing the Square
Question: Write \( f(x) = x^2 - 6x + 11 \) in vertex form and state the minimum value.
Solution
\( f(x) = (x^2 - 6x) + 11 \)
\( = (x^2 - 6x + 9) + 11 - 9 \)
\( = (x - 3)^2 + 2 \)
Vertex: \( (3, 2) \), \( a = 1 > 0 \) → minimum value is \( 2 \).
Example 3: Graph Transformations
Question: Describe the transformation from \( y = x^2 \) to \( y = -2(x+3)^2 - 1 \).
Solution
Starting from \( y = x^2 \):
- Shift left by 3 units (\( h = -3 \))
- Vertical stretch by factor 2
- Reflect in x-axis (because \( a = -2 \))
- Shift down by 1 unit
Answer: Left 3, stretch by 2, reflect, down 1.
8. DSE-Style Practice Questions
Section A(2) & Section B Style
Question 1 MC
What is the vertex of \( f(x) = 3(x-4)^2 + 5 \)?
A. \( (4, 5) \) B. \( (-4, 5) \) C. \( (4, -5) \) D. \( (-4, -5) \)
Question 2 MC
What is the axis of symmetry of \( f(x) = -2x^2 + 8x - 3 \)?
A. \( x = -2 \) B. \( x = 2 \) C. \( x = 4 \) D. \( x = -4 \)
Question 3 Short Answer
Write \( f(x) = x^2 + 8x + 15 \) in vertex form.
Question 4 Short Answer
Find the minimum value of \( f(x) = 2x^2 - 12x + 5 \).
Question 5 MC
Which transformation from \( y = x^2 \) gives \( y = (x+2)^2 - 3 \)?
A. Left 2, up 3 B. Left 2, down 3 C. Right 2, up 3 D. Right 2, down 3
9. Solutions with Explanations
Question 1: A. \( (4, 5) \)
Vertex form \( a(x-h)^2 + k \) gives vertex \( (h,k) = (4,5) \).
Question 2: B. \( x = 2 \)
\( h = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2 \).
Question 3: \( (x+4)^2 - 1 \)
\( x^2 + 8x + 15 = (x^2 + 8x + 16) - 1 = (x+4)^2 - 1 \).
Question 4: \( -13 \)
\( f(x) = 2(x^2 - 6x) + 5 = 2[(x-3)^2 - 9] + 5 = 2(x-3)^2 - 18 + 5 = 2(x-3)^2 - 13 \)
Minimum value = \( -13 \).
Question 5: B. Left 2, down 3
\( y = (x+2)^2 - 3 \) means shift left by 2 and down by 3.
9. Exercise
Click the following link to have
An Exercise on Quadratic Functions | Graphs, Vertex and Axis of Symmetry
Key Takeaways
What You Should Remember
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Vertex form: \( f(x) = a(x-h)^2 + k \), vertex \( (h,k) \), axis \( x = h \).
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General form: \( f(x) = ax^2 + bx + c \), vertex \( h = -\frac{b}{2a} \), \( k = f(h) \).
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Completing the square converts general form to vertex form.
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Opening direction: \( a > 0 \) → upward (minimum), \( a < 0 \) → downward (maximum).
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Transformations: \( a \) affects stretch/reflection, \( h \) horizontal shift, \( k \) vertical shift.
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This topic guarantees 4–6 marks in DSE Paper 1 Section A(2) and Section B – master these skills!
Summary Checklist for Revision
- Vertex form: \( a(x-h)^2 + k \)
- General form: \( ax^2 + bx + c \)
- Vertex: \( (h,k) \) or \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \)
- Axis of symmetry: \( x = h \)
- Completing the square: convert general to vertex
- Max/min: \( a > 0 \) min, \( a < 0 \) max
- Sketching: vertex, axis, intercepts, opening
- Transformations: translations, stretches, reflections