Plotting Quadratic Graphs and Their Roots

What a quadratic graph is

A quadratic is an equation whose highest power of x is x², such as:

y = x² + 2x − 3

When you plot it, the points form a smooth, symmetrical U-shape called a parabola. This is very different from the straight lines you met in straight-line graphs and gradients — a quadratic curves.

Step 1: build a table of values

The reliable way to plot any quadratic is a table of values. Choose a range of x-values (the question usually gives one), substitute each into the equation, and record y.

The one rule to be careful with: square the x first, before adding the other terms, and remember a negative squared is positive.

Worked example 1: y = x²

Plot y = x² for x from −3 to 3.

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

Notice how the y-values are symmetrical about x = 0. Plot the seven points (−3, 9), (−2, 4), …, (3, 9) and join them with one smooth curve.

y
9 ●               ●
  |  \           /
4 |   ●         ●
  |    \       /
1 |     ●     ●
0 |_______●_________ x
   -3 -2 -1 0 1 2 3

• The lowest point (0, 0) is the turning point (here the minimum).
• The vertical line x = 0 is the line of symmetry — the two halves mirror each other.

Step 2: read the roots

The roots of a quadratic are the x-values where the curve crosses the x-axis, that is where y = 0.

For y = x² the curve only touches the x-axis at (0, 0), so there is a single (repeated) root at x = 0.

Worked example 2: a curve with two roots

Plot y = x² − 4 for x from −3 to 3, and find its roots.

Work out each value carefully (square first, then subtract 4):

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

Plot and join smoothly. The curve crosses the x-axis at x = −2 and x = 2, so the roots are x = −2 and x = 2. The turning point is (0, −4) and the line of symmetry is x = 0.

You can confirm the roots algebraically: x² − 4 = 0 → x² = 4 → x = ±2.

Worked example 3: a quadratic with an x-term

Plot y = x² + 2x − 3 for x from −4 to 2.

Substitute each x. For example at x = −4: (−4)² + 2(−4) − 3 = 16 − 8 − 3 = 5.

x	−4	−3	−2	−1	0	1	2
x²	16	9	4	1	0	1	4
+2x	−8	−6	−4	−2	0	2	4
−3	−3	−3	−3	−3	−3	−3	−3
y	5	0	−3	−4	−3	0	5

The curve crosses the x-axis at x = −3 and x = 1, so the roots are −3 and 1.

• Turning point: the lowest y is −4, at x = −1, so the minimum is (−1, −4).
• Line of symmetry: x = −1 (exactly halfway between the two roots).

Check the roots by factorising: x² + 2x − 3 = (x + 3)(x − 1), which is 0 when x = −3 or x = 1. ✓ This links directly to quadratic equations.

Reading the graph: a checklist

From a plotted parabola you can read:

Feature	How to find it
Roots	x-values where the curve crosses the x-axis (y = 0)
y-intercept	the y-value where x = 0
Turning point	the lowest (or highest) point on the curve
Line of symmetry	the vertical line through the turning point

Three possibilities for roots

A parabola can meet the x-axis in three ways:

• Two roots — crosses the axis twice (e.g. y = x² − 4).
• One root — just touches the axis (e.g. y = x²).
• No real roots — never reaches the axis (e.g. y = x² + 1, which stays above it).

A reliable method

1. Make a table of values across the given range of x.
2. Square each x carefully (negatives become positive), then apply the rest of the equation.
3. Plot every point on the grid.
4. Join them with one smooth curve, not straight segments.
5. Read off the roots (x-axis crossings), the turning point, and the line of symmetry.

Activity: plot and read

For each, complete a table from x = −3 to 3, plot the curve, then state the roots.

1. y = x² − 1
2. y = x² − 2x
3. y = x² + 1

Answers:

1. y-values: 8, 3, 0, −1, 0, 3, 8. Crosses at x = −1 and x = 1 → roots −1 and 1; turning point (0, −1).
2. y-values: 15, 8, 3, 0, −1, 0, 3 (for x = −3…3). Crosses at x = 0 and x = 2 → roots 0 and 2; turning point (1, −1).
3. y-values: 10, 5, 2, 1, 2, 5, 10. Curve never reaches the x-axis → no real roots; turning point (0, 1).

Where this leads

Plotting reveals what the algebra means: the roots you read off the x-axis are exactly the solutions of quadratic equations, and the same plotting skill lets you solve equations graphically by finding where two graphs meet.