Quadratic Graphs and the Parabola

Completing a table of values, drawing a smooth quadratic curve, reading roots and the line of symmetry, and sketching from completed square form.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 53 real past-paper questions below

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The Shape of a Quadratic

Every graph of $y = a x^{2} + b x + c$ is a smooth U, or an upside-down U if a is negative. It is symmetrical.

Rule

The lowest (or highest) point is the turning point. The mirror line through it is the line of symmetry.

Key idea

Most questions here start the same way: complete the table (2–3 marks), then draw the curve (2–4 marks). Then 1-mark parts ask for the line of symmetry or the lowest point.

Key idea

The name parabola comes from the Greek geometer Apollonius of Perga, around 225 BC. In his books on curves, the Conics, he named the parabola, the ellipse and the hyperbola. He is believed to have studied in Alexandria, in Egypt.

Completing the Table

Exams give a table with gaps. Put each x into the equation, with negatives in brackets, and do the powers first.

Worked example

1
$y = x^{2} + 3 x - 2$ — The curve. We need the y value when x = −4.
2
$y = (- 4)^{2} + 3 \times (- 4) - 2$ — Replace every x with (−4). The brackets keep the minus attached to the 4.
3
$(- 4)^{2} = 16$ — Do the power first. Minus times minus is plus, so the square is +16.
4
$3 \times (- 4) = - 12$ — Next the multiplication. Plus times minus is minus.
5
$y = 16 - 12 - 2 = 2$ — Now add along the line. So the table gets y = 2 under x = −4.

When x² has a minus in front. Many questions look like $y = 2 + 4 x - x^{2}$ . The minus belongs outside the square.

Worked example

1
$y = 2 + 4 \times (- 2) - (- 2)^{2}$ — Put x = −2 in, in brackets, everywhere you see x.
2
$(- 2)^{2} = 4$ — Square the bracket first. The minus in front of $x^{2}$ is not squared, so it waits.
3
$- (- 2)^{2} = - 4$ — Now apply the waiting minus: minus 4. This is the line most students get wrong.
4
$4 \times (- 2) = - 8$ — The middle term: plus times minus is minus.
5
$y = 2 - 8 - 4 = - 10$ — Add along the line. So at x = −2, y = −10.

Common mistake

Writing −(−2)² as +4. Square the bracket first, which gives 4, then put the minus in front: −4.

Key idea

If the equation has a fraction, such as $\frac{x ^{2}}{2}$ , square first, then divide: at x = 3 it is 9 ÷ 2 = 4.5.

Drawing the Curve

Plot every point from your table, then join them with one smooth curve. Do not use a ruler.

Worked example

1
$(- 5, 8), (- 4, 2), (- 3, - 2), (- 2, - 4)$ — Plot the left half of the table for $y = x^{2} + 3 x - 2$ . Each x is across, each y is up or down.
2
$(- 1, - 4), (0, - 2), (1, 2), (2, 8)$ — Plot the right half. The y values come back in reverse order. That is the symmetry, and a quick check on your table.
3
$(- 1.5, - 4.25)$ — Between the two −4s the curve dips a little lower, to −4.25. The card after the graph shows how to find this point.
4
$one smooth curve through all 8 points$ — Draw it in one sweep, without lifting the pencil. The bottom must be round, not flat or pointed.

Common mistake

A flat line or a sharp point between (−2, −4) and (−1, −4). Examiners want the curve to dip below −4 there.

Key idea

Most of the drawing marks are for correctly plotted points, and they follow your own table. So plot every value you worked out, even one you are unsure of.

Line of Symmetry and Turning Point

Many questions ask for the line of symmetry and the lowest (or highest) point. Your table gives you both.

Worked example

1
$x = - 2 : y = - 4 x = - 1 : y = - 4$ — In the table for $y = x^{2} + 3 x - 2$ , two x values give the same y. The mirror line is exactly between them.
2
$\frac{( - 2 ) + ( - 1 )}{2} = \frac{- 3}{2} = - 1.5$ — Halfway between two numbers: add them, then halve.
3
$x = - 1.5$ — Write it as an equation. The line is upright, and every point on it has x = −1.5. Writing just −1.5 can lose the mark.
4
$y = (- 1.5)^{2} + 3 \times (- 1.5) - 2$ — The turning point sits on the mirror line, so put x = −1.5 into the equation, in brackets.
5
$y = 2.25 - 4.5 - 2 = - 4.25$ — (−1.5)² = 2.25 and 3 × (−1.5) = −4.5. Then add along.
6
$(- 1.5, - 4.25)$ — The lowest point, because the curve is a U. Check it against your drawing.

Shortcut. This formula gives the line of symmetry straight from the equation:

Formula

x = - \frac{b}{2 a}

$a$ is the number in front of $x^{2}$ and $b$ is the number in front of $x$ , each with its sign. The constant term is not used.

Worked example

1
$y = 2 + 4 x - x^{2}$ — Read the numbers: a = −1 (the minus in front of x²) and b = 4.
2
$x = - \frac{4}{2 \times ( - 1 )}$ — Put them in the formula. Keep −1 in brackets so its minus is not lost.
3
$x = - \frac{4}{- 2} = 2$ — 2 × (−1) = −2, and 4 ÷ (−2) = −2. The minus in front of the fraction makes it +2. So the line of symmetry is x = 2.
4
$y = 2 + 4 \times 2 - 2^{2}$ — Put x = 2 back into the equation to find the y value of the turning point.
5
$y = 2 + 8 - 4 = 6$ — 4 × 2 = 8 and 2² = 4. So the turning point is (2, 6).
6
$(2, 6)$ — This is the highest point, because the minus in front of x² turns the curve upside down.

Using the Symmetry

Some questions give a point far off the grid and ask for its partner with the same y. Use the line of symmetry, not the graph.

Worked example

1
$(5, 38) and (t, 38)$ — Both points are on $y = x^{2} + 3 x - 2$ . Same y means they are mirror images in the line x = −1.5.
2
$5 - (- 1.5) = 5 + 1.5 = 6.5$ — How far is x = 5 from the mirror line? Subtract. Minus a minus is plus, so the gap is 6.5.
3
$t = - 1.5 - 6.5 = - 8$ — Go the same distance, 6.5, to the other side of the line. So t = −8.
4
$(- 8)^{2} + 3 \times (- 8) - 2 = 64 - 24 - 2 = 38$ — Check by putting x = −8 into the equation. It gives 38, so the answer is right.

Memory trick

Same y, same distance from the mirror line.

Key idea

The two places where the curve crosses the x-axis are mirror images too. If you know one, you can find the other the same way.

Reading the Graph

Once the curve is drawn, you can read answers straight off it.

Worked example

1
$x^{2} + 3 x - 2 = 0$ — This asks where y = 0. On the graph, y = 0 is the x-axis.
2
$x \approx - 3.6 or x \approx 0.6$ — Find the two places the curve crosses the x-axis and read the x values, to 1 decimal place.
3
$\frac{- 3.6 + 0.6}{2} = - 1.5$ — Check: the two crossings should sit either side of the line of symmetry, x = −1.5. They do.

Key idea

To solve $x^{2} + 3 x - 2 = 4$ , draw the flat line y = 4 and read where it meets the curve. The lesson Solving Quadratics from a Graph shows this step by step.

Sketching a Quadratic (Extended/Higher)

A sketch is not plotted. It shows the shape and labels the points where the curve meets each axis, plus the turning point.

Worked example

1
$y = x^{2} - 2 x - 8$ — The question. The $x^{2}$ is positive, so the curve is a U shape.
2
$x = 0 : y = 0 - 0 - 8 = - 8$ — Put x = 0 to find where it crosses the y-axis. Every x term becomes 0, leaving the number on its own: −8.
3
$x^{2} - 2 x - 8 = 0$ — It crosses the x-axis where y = 0, so set the equation equal to 0.
4
$(x - 4) (x + 2) = 0$ — Factorise: −4 and +2 multiply to −8 and add to −2.
5
$x = 4 or x = - 2$ — A product is 0 only if one bracket is 0. So it crosses the x-axis at 4 and −2.
6
$x = \frac{- 2 + 4}{2} = 1$ — The turning point is halfway between the two crossings.
7
$y = 1^{2} - 2 \times 1 - 8 = - 9$ — Put x = 1 in to get its y value. So the lowest point is (1, −9).

Key idea

Label −8, −2, 4 and (1, −9) on the sketch. Each labelled value earns a mark. If the equation is already in completed square form, such as $y = 4 (x - 1)^{2} - 9$ , read the turning point (1, −9) straight off. See the lesson Completed Square Form and Turning Points.

From a sketch back to the equation. Some questions show a U crossing the x-axis at two numbers and ask for the equation $y = x^{2} + p x + q$ .

Worked example

1
$x = - 1 and x = 5$ — Read the two crossing points off the sketch.
2
$y = (x + 1) (x - 5)$ — Each crossing gives a bracket with the opposite sign: x + 1 is 0 when x = −1, and x − 5 is 0 when x = 5.
3
$y = x^{2} - 5 x + x - 5$ — Expand: multiply each term in the first bracket by each term in the second.
4
$y = x^{2} - 4 x - 5$ — Collect the x terms: −5x + x = −4x. So p = −4 and q = −5.
5
$5^{2} - 4 \times 5 - 5 = 25 - 20 - 5 = 0$ — Check with x = 5. The answer is 0, so the curve does cross at 5.

Which Method? Read the Question

Table first, then the curve. Then each small part has its own move.

You see	You do
“Complete the table”	Substitute, negatives in brackets, powers first
“Draw the graph”	Plot all points, one smooth curve
“Line of symmetry”	Halfway between equal y values; write x = …
“Lowest / highest point”	Put the symmetry line's x into the equation
“Use symmetry to find”	Same distance on the other side of the line
“Use your graph to solve”	Read where the curve crosses the line
“Sketch”	Right shape, both axes and turning point labelled

Key idea

Use the symmetry to check your table: matching y values sit either side of the middle.

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