Relations, Functions and Graphs: Quadratic functions — graphs, vertex, axis of symmetry and interpretation

What you'll learn

Quadratic functions form a significant portion of the CXC CSEC Mathematics syllabus under Relations, Functions and Graphs. You must understand how to sketch parabolas, identify key features including the vertex and axis of symmetry, and interpret these graphs in practical contexts. Exam questions test both algebraic manipulation and graphical interpretation skills, often worth 8-12 marks in Paper 2.

Key terms and definitions

Quadratic function — A function of the form f(x) = ax² + bx + c where a, b, and c are constants and a ≠ 0.

Parabola — The U-shaped curve that forms the graph of a quadratic function; opens upward when a > 0 and downward when a < 0.

Vertex — The turning point of a parabola; the maximum point when the parabola opens downward or the minimum point when it opens upward.

Axis of symmetry — The vertical line passing through the vertex that divides the parabola into two mirror images; equation is x = h where h is the x-coordinate of the vertex.

Roots (or zeros) — The x-values where the parabola crosses the x-axis; solutions to the equation ax² + bx + c = 0.

y-intercept — The point where the parabola crosses the y-axis; always at (0, c) for the function f(x) = ax² + bx + c.

Discriminant — The expression b² - 4ac that determines the number and nature of roots; positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Turning point form — The expression f(x) = a(x - h)² + k where (h, k) is the vertex of the parabola.

Core concepts

Standard form and general form of quadratic functions

The general form of a quadratic function is f(x) = ax² + bx + c where:

• a determines the shape and direction (a > 0 opens upward, a < 0 opens downward)
• b affects the position of the axis of symmetry
• c is the y-intercept

The turning point form or vertex form is f(x) = a(x - h)² + k where (h, k) represents the vertex coordinates. CXC CSEC Mathematics examiners frequently test conversion between these forms using the completing-the-square method.

Finding the vertex and axis of symmetry

For a quadratic function in general form f(x) = ax² + bx + c:

Method 1: Using the formula

• The x-coordinate of the vertex is x = -b/(2a)
• Substitute this x-value back into the function to find the y-coordinate
• The axis of symmetry is the line x = -b/(2a)

Method 2: Completing the square

• Rewrite f(x) = ax² + bx + c in the form f(x) = a(x - h)² + k
• The vertex is (h, k)
• The axis of symmetry is x = h

Example: For f(x) = 2x² - 8x + 5

Method 1: x = -(-8)/(2×2) = 8/4 = 2 f(2) = 2(2)² - 8(2) + 5 = 8 - 16 + 5 = -3 Vertex: (2, -3), Axis of symmetry: x = 2

Method 2: f(x) = 2(x² - 4x) + 5 = 2(x² - 4x + 4 - 4) + 5 = 2(x - 2)² - 8 + 5 = 2(x - 2)² - 3 Vertex: (2, -3), Axis of symmetry: x = 2

Sketching quadratic graphs

When sketching parabolas for CXC CSEC Mathematics examinations, include these features:

1. Determine the shape: Check the sign of a (positive = U-shape, negative = ∩-shape)

2. Find the y-intercept: Set x = 0, giving the point (0, c)

3. Calculate the vertex: Use x = -b/(2a) method

4. Find the x-intercepts (if they exist): Solve ax² + bx + c = 0 using factorization, the quadratic formula, or completing the square

5. Draw the axis of symmetry: Vertical dashed line through the vertex

6. Plot additional points: Choose x-values on either side of the axis of symmetry for accuracy

7. Draw a smooth curve: Connect all points with a smooth parabola, ensuring symmetry about the axis

Key features and properties

Domain and range:

• Domain of any quadratic function: all real numbers (x ∈ ℝ)
• Range depends on the vertex:
  • If a > 0 (opens upward): f(x) ≥ k where k is the y-coordinate of the minimum point
  • If a < 0 (opens downward): f(x) ≤ k where k is the y-coordinate of the maximum point

Line of symmetry properties:

• Any two points equidistant from the axis of symmetry have the same y-coordinate
• If x₁ and x₂ are the roots, the axis of symmetry is x = (x₁ + x₂)/2

Interpreting quadratic graphs in context

CXC CSEC Mathematics papers often present quadratic functions modeling real-world situations. Common Caribbean contexts include:

Projectile motion: A cricket ball hit at Sabina Park in Jamaica or a football kicked in Queen's Park Savannah, Trinidad, follows a parabolic path. The vertex represents maximum height, and the x-intercepts show where the ball lands.

Business applications: Profit functions for a Jamaican patty vendor or a Barbadian rum distillery often follow quadratic patterns. The vertex indicates maximum profit, while roots show break-even points.

Agricultural yield: Crop production in relation to fertilizer application may follow a quadratic model, with the vertex showing optimal fertilizer quantity.

When interpreting:

• Identify what the axes represent (include units)
• State the meaning of the vertex in context
• Explain intercepts in practical terms
• Consider realistic domain restrictions (e.g., time cannot be negative)

Using technology and tables of values

For CXC CSEC Mathematics, you may need to:

Create a table of values:

• Choose at least 5 x-values centered around the vertex
• Calculate corresponding y-values
• Plot these coordinate pairs
• Draw a smooth curve through the points

Read from graphs:

• Estimate coordinates of key points
• Determine maximum or minimum values
• Find approximate solutions to equations
• Identify intervals where the function is increasing or decreasing

Worked examples

Example 1: Complete analysis of a quadratic function

Question: The function f(x) = -x² + 6x - 5 models the height in meters of a fountain jet at a beach resort in Barbados, where x represents horizontal distance in meters from the fountain's base.

(a) Express f(x) in the form f(x) = -(x - h)² + k. (3 marks) (b) State the coordinates of the vertex and the equation of the axis of symmetry. (2 marks) (c) Find the maximum height reached by the water. (1 mark) (d) Calculate the horizontal distance between the two points where the water hits the ground. (3 marks)

Solution:

(a) f(x) = -x² + 6x - 5 = -(x² - 6x) - 5 = -(x² - 6x + 9 - 9) - 5 = -(x² - 6x + 9) + 9 - 5 = -(x - 3)² + 4 ✓✓✓

(b) Vertex: (3, 4) ✓ Axis of symmetry: x = 3 ✓

(c) Maximum height = 4 meters ✓

(d) When water hits ground, f(x) = 0 -x² + 6x - 5 = 0 x² - 6x + 5 = 0 (x - 1)(x - 5) = 0 ✓ x = 1 or x = 5 ✓ Horizontal distance = 5 - 1 = 4 meters ✓

Example 2: Sketching from vertex form

Question: A mango farmer in Trinidad finds that his daily profit P (in dollars) from selling x kilograms of mangoes is given by P(x) = -2(x - 15)² + 450.

(a) Sketch the graph of P(x), showing the vertex and intercepts. (4 marks) (b) State the maximum profit and the quantity that produces it. (2 marks) (c) Determine the range of quantities that result in a profit greater than $400. (3 marks)

Solution:

(a) Vertex form shows vertex at (15, 450) ✓ P opens downward (a = -2 < 0) ✓ P-intercept: P(0) = -2(0-15)² + 450 = -450 + 450 = 0 ✓ x-intercepts: Set P(x) = 0 -2(x - 15)² + 450 = 0 (x - 15)² = 225 x - 15 = ±15 x = 30 or x = 0 ✓ [Graph should show downward parabola with vertex at (15, 450) and intercepts at (0, 0) and (30, 0)]

(b) Maximum profit = $450 ✓ Quantity = 15 kg ✓

(c) -2(x - 15)² + 450 > 400 -2(x - 15)² > -50 (x - 15)² < 25 ✓ -5 < x - 15 < 5 10 < x < 20 ✓ The farmer should sell between 10 kg and 20 kg ✓

Example 3: Finding a quadratic function from key features

Question: A parabola has its vertex at (4, -2) and passes through the point (6, 6). Find the equation of the parabola in: (a) Vertex form (2 marks) (b) General form (2 marks)

Solution:

(a) Vertex form: f(x) = a(x - 4)² - 2 Substitute (6, 6): 6 = a(6 - 4)² - 2 6 = 4a - 2 8 = 4a a = 2 ✓ f(x) = 2(x - 4)² - 2 ✓

(b) f(x) = 2(x - 4)² - 2 = 2(x² - 8x + 16) - 2 = 2x² - 16x + 32 - 2 ✓ = 2x² - 16x + 30 ✓

Common mistakes and how to avoid them

Mistake: Confusing the vertex formula — Students write x = -b/2a instead of x = -b/(2a), forgetting the brackets around 2a. Always write the denominator as (2a) or use x = -b ÷ (2a) to avoid order-of-operations errors.

Mistake: Sign errors when completing the square — When factoring out a negative coefficient, students forget to apply it to the constant added and subtracted inside the bracket. For f(x) = -x² + 6x - 5, write -(x² - 6x) - 5, then -(x² - 6x + 9) + 9 - 5, carrying the negative through correctly.

Mistake: Misidentifying the vertex from vertex form — In f(x) = a(x - h)² + k, the vertex is (h, k), not (-h, k). For f(x) = 2(x - 3)² + 5, the vertex is (3, 5), not (-3, 5). Remember: the sign changes because of the subtraction.

Mistake: Assuming all parabolas have x-intercepts — When b² - 4ac < 0, the parabola does not cross the x-axis. Check the discriminant before attempting to find roots by factorization or the quadratic formula.

Mistake: Incorrect range notation — For a minimum point at y = k with a > 0, write f(x) ≥ k or y ≥ k, not f(x) > k. The vertex value is included in the range. Use "or equal to" unless the context explicitly excludes it.

Mistake: Poor graph sketching — Students draw pointed vertices instead of smooth curves, or forget to show symmetry. Practice drawing smooth parabolas, and always check that points equidistant from the axis of symmetry have equal y-values.

Exam technique for Relations, Functions and Graphs: Quadratic functions

Command words and response strategies:

• "Sketch" requires a labelled diagram showing key features (vertex, intercepts, axis of symmetry) but not precise plotting of many points
• "Plot" or "draw accurately" requires using graph paper with a table of values and careful point-by-point plotting
• "Find" or "determine" the vertex requires showing algebraic working, not just reading from a graph
• "State" the maximum/minimum can often be answered directly from vertex form without additional working

Mark allocation patterns:

• Completing the square: typically 3 marks (1 for correctly forming the square, 1 for simplifying, 1 for final form)
• Finding vertex: 2-3 marks (1-2 for x-coordinate using formula or method, 1 for y-coordinate)
• Sketching graphs: 4-5 marks (1 mark each for shape, vertex, y-intercept, x-intercepts, axis of symmetry)
• Contextual interpretation: 2-3 marks per part (1 mark for calculation, 1-2 marks for explanation in context)

Structuring multi-part answers: Label parts (a), (b), (c) clearly. Show all algebraic steps even when using the calculator. For graphs, use a ruler for the axis of symmetry and plot at least one point on each side of the vertex for accuracy.

Time management: Allocate approximately 2 minutes per mark. A 9-mark quadratic function question should take about 18 minutes. If stuck on finding roots algebraically, use the quadratic formula as your reliable method.

Quick revision summary

Quadratic functions f(x) = ax² + bx + c produce parabolic graphs. The vertex is found using x = -b/(2a) or by completing the square to form f(x) = a(x - h)² + k where vertex is (h, k). The axis of symmetry is the vertical line x = h. When a > 0, the parabola opens upward with minimum at the vertex; when a < 0, it opens downward with maximum at the vertex. The y-intercept is always c, and x-intercepts exist only when b² - 4ac ≥ 0. Master both algebraic methods and graphical interpretation for CXC CSEC success.