🪃 Quadratic Graphs

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

Plot quadratic graphs using a table of values
Identify the vertex (turning point) and axis of symmetry
Understand the effect of a, b, c in y = ax² + bx + c
Read roots (x-intercepts) from a graph
Use graphs to solve quadratic equations approximately

📖 Learn

1. The Parabola

A quadratic function produces a U-shaped or ∩-shaped curve called a parabola.

y = ax² + bx + c

a > 0: U-shape (opens upward) — minimum turning point

a < 0: ∩-shape (opens downward) — maximum turning point

c: the y-intercept (substitute x = 0)

💡 The bigger |a|, the narrower the parabola. a = 1 gives a standard width.

2. Table of Values

To plot a quadratic, create a table of x and y values, then plot and join with a smooth curve.

Example: y = x² − 2x − 3, for x from −2 to 4

Substitute each x: at x=−2: (−2)²−2(−2)−3 = 4+4−3 = 5

Continue for all x values, plot each (x, y) point

Join with a smooth curved line — do NOT join with straight line segments!

⚠️ Always draw a smooth curve through ALL plotted points. Straight lines give wrong shape.

3. Key Features

A parabola has several important features:

Roots (x-intercepts): Where y = 0 — where the curve crosses the x-axis. A quadratic can have 0, 1, or 2 roots.

Vertex (turning point): The minimum (U) or maximum (∩) point.

Axis of symmetry: Vertical line through the vertex. x = −b/(2a) for y = ax²+bx+c

y-intercept: Where x = 0, so y = c

💡 The vertex x-coordinate = average of the two roots (if they exist).

4. Solving with a Graph

The roots of y = ax²+bx+c are the x-values where the curve crosses the x-axis (y = 0).

To solve ax²+bx+c = k: draw y = k on the same axes, read off x-values at intersections

Example: Solve x²−2x−3 = 0 → read where curve crosses y = 0 → x = −1 and x = 3

Approximate solutions: graphical solutions may not be exact — state to 1 d.p. if needed

5. Transformations of y = x²

y = x² + k: shifts the graph k units up (k > 0) or down (k < 0)

y = (x − h)²: shifts the graph h units to the right

y = −x²: reflects in the x-axis (flips upside down)

y = 2x²: makes the parabola narrower

y = ½x²: makes the parabola wider

💡 Vertex of y = (x−h)² + k is at (h, k).

✏️ Worked Examples

Example 1 – Table of Values

Complete the table for y = x² − 4 for x = −3 to 3.

x	−3	−2	−1	0	1	2	3
x²	9	4	1	0	1	4	9
y = x²−4	5	0	−3	−4	−3	0	5

Roots: y = 0 at x = −2 and x = 2

Vertex: minimum at (0, −4) — the lowest point

Axis of symmetry: x = 0

Example 2 – Reading Features from a Graph

y = x² − 2x − 3. Identify roots, vertex, axis of symmetry.

y-intercept: x = 0 → y = −3

Roots: factorise (x−3)(x+1) = 0 → x = 3 or x = −1

Axis of symmetry: x = (3 + (−1))/2 = 1 OR use x = −b/2a = 2/2 = 1

Vertex y-value: y = 1² − 2(1) − 3 = −4 → vertex = (1, −4)

Example 3 – Solving Graphically

Use the graph of y = x² − x − 6 to solve x² − x − 6 = 0.

Method: Draw the curve. Read the x-values where it crosses y = 0.

Roots: The curve crosses at x = −2 and x = 3.

Check: (−2)² − (−2) − 6 = 4+2−6 = 0 ✓ 3²−3−6 = 9−3−6 = 0 ✓

Example 4 – Sketching with Key Features

Sketch y = −x² + 4, labelling intercepts and vertex.

Shape: a < 0 (a = −1) → ∩-shape (maximum)

y-intercept: x=0 → y = 4, point (0, 4)

Roots: 0 = −x²+4 → x² = 4 → x = ±2, points (−2, 0) and (2, 0)

Vertex: Maximum at (0, 4) — the axis of symmetry is x = 0

🎨 Visualizer

🪃 Parabola Plotter

Enter a, b, c for y = ax² + bx + c.

a: b: c:

📊 Table of Values Generator

Generates y values for y = ax² + bx + c over x = −5 to 5.

a: b: c:

❓ Identify the Features Quiz

Exercise 1 – Table of Values for y = x²

y = x². Fill in the y-value for each x.

Exercise 2 – Table of Values for y = x² − 4

y = x² − 4. Find y for each x value.

Exercise 3 – Roots and y-intercepts

Find the y-intercept (substitute x=0) or roots (set y=0) as directed.

Exercise 4 – Vertex and Axis of Symmetry

Find the x-coordinate of the axis of symmetry using x = −b/(2a), or the vertex y-value.