A quadratic function (intro) · Lập trình với Martin

A function that is not a straight line

So far every function drew a straight line. Today you meet the first one that does not: a quadratic function. The simplest of them is y = x².

That little number on top means a square: x² = x · x. For x = 3, y = 9. For x = 5, y = 25. Nothing new — you know powers.

What is new is what it does to the graph. Try substituting a negative number: for x = −3, y = (−3) · (−3) = +9. The same as for +3! A negative input gives the same output as a positive one.

So the graph is not a straight line, but an arch in the shape of the letter U. It is called a parabola. It is perfectly symmetric about the vertical axis — the left half is a mirror of the right.

A parabola is drawn by the world around you all the time: a ball thrown at an angle flies on a parabola (just upside down), a jet of water from a fountain too. Today you will draw it from a table and learn to recognise it.

Adam is shooting at a hoop and Sofie films him in slow motion. “Look at the ball’s path,” she plays the clip, “up, the peak, down — a beautiful arch.” Adam traces the screen with a finger: “And it is symmetric. The ball rises for as long as it then falls.” Sofie nods: “That is a parabola. The same curve as the graph of y = x², just flipped. Physics draws it on its own — in maths we learn to read it.”

💡

For a quadratic function, substitute negative x too — otherwise you only draw half a parabola and the shape gets away. A table from −3 to +3 is the base.

A parabola from a table

Step 1: A table for y = x², including negative x.

x: −3, −2, −1, 0, 1, 2, 3 y: 9, 4, 1, 0, 1, 4, 9

Step 2: Read what the table says. The values fall towards zero and then grow as a mirror. The lowest point is [0; 0] — it is called the vertex of the parabola. And y is never negative: a square cannot come out below zero.

Step 3: Plot and join with a smooth arch. Put the ruler down here! A parabola is a curve — join the points with a smooth hand, no broken lines between points.

Symmetry as a check: y for x = −2 must be the same as for x = 2. If the table does not have the same numbers mirrored, there is a counting mistake.

Variants you will meet:

y = 2x² — the same shape, but thinner (it grows twice as fast). y = x² + 1 — the whole parabola shifted 1 up, vertex at [0; 1]. y = −x² — flipped upside down, like a ball’s path. The vertex is the highest point.

A recognition test at the end: in the rule of a linear function, x stands on its own. As soon as you see x², do not expect a straight line — it will be an arch.

Example 1: table and graph of y = x²

Question: Build a table of y = x² for x from −3 to 3 and draw the graph.

Method: I count one by one: (−3)² = 9, (−2)² = 4, (−1)² = 1, 0² = 0, 1² = 1, 2² = 4, 3² = 9.

I plot seven points and join them with a smooth arch. The vertex is at the origin [0; 0].

Symmetry check: the pair x = −3 and x = 3 both give y = 9. The left half mirrors the right. It matches.

Example 2: a shifted parabola y = x² + 2

Question: How does the graph of y = x² + 2 differ from the graph of y = x²?

Method: I calculate a few values: for x = 0, y = 2. For x = 1, y = 3. For x = 2, y = 6. For x = −1, y = 3.

Each value is exactly 2 bigger than for y = x². So the whole arch has lifted 2 up.

Answer: The same shape, the vertex has moved from [0; 0] to [0; 2]. The added number after x² shifts the parabola vertically — exactly like b shifted a straight line.

Example 3: a straight line, or a parabola?

Question: Decide from the rule what is a straight line and what is a parabola: y = 3x + 7, y = x² − 4, y = −2x, y = −x² + 5.

Method: I look for x².

y = 3x + 7: only x, no power → a straight line (it rises). y = x² − 4: there is x² → a parabola, shifted 4 down, vertex [0; −4]. y = −2x: only x → a straight line (it falls, goes through the origin). y = −x² + 5: x² with a minus → a parabola upside down, vertex [0; 5] is the highest point.

Answer: straight line, parabola, straight line, flipped parabola. One thing decides: is there x² in the rule?

⚠️

(−3)² is not −9. Minus times minus is plus, so (−3)² = 9. Without a bracket it is different: −3² is read “minus (3²)” and that is −9. When you substitute negative numbers into x², always write a bracket — one pair of brackets decides the sign of the whole result.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Practise substituting into expressions — with parabolas you substitute all the time, just with a power extra. Write a whole number.

Practise the questions

On paper

Two parabolas and one recognition test. Tham gia points with an arch, not a ruler.

Build a table of y = x² for x from −3 to 3 and draw the parabola. Check the symmetry of the table.
On the same picture draw y = x² − 2. Where has the vertex moved?
Calculate y = 2x² for x = −2, −1, 0, 1, 2 and compare with y = x²: which parabola is thinner?
Sort into straight lines and parabolas: y = 5x − 1, y = x² + 3, y = −x², y = 0.5x. For parabolas write the vertex.

Now you 💪

You know why (−3)² = 9, and when substituting you write brackets.
Your table of y = x² is mirror-symmetric and the graph is a smooth arch with a vertex.
From the rule you can tell a straight line from a parabola at a glance.

✅

Done when: You have drawn two parabolas with the right vertex, the tables are symmetric and the straight-line/parabola recognition is without a mistake.

What to take from this lesson

A quadratic function contains x². Its graph is a parabola, not a straight line.
A parabola is symmetric — negative x give the same y as positive ones.
The lowest (for a flipped one, the highest) point of a parabola is the vertex.
An added number shifts the parabola up/down, a minus in front of x² flips it.
When substituting negative numbers into a power, always write brackets.