SumNinja

Recognise a quadratic rule and its curve

About 3 min

A square rule gives matching heights on either side of its middle. Use that pattern and the turn to check the graph.

A game map describes a valley with y = x² − 1. Here x gives left-right position and y gives height. The little 2 means x times itself. Find the lowest point, then check a matching pair of heights to choose the right graph.

Find the valley floor and two matching sides

Read the graph of y = x² − 1.

  1. Check the middle of this valley

    x=0⇒y=02−1=−1x=0\Rightarrow y=0^2-1=-1

    Squaring never gives a negative result. Zero gives the smallest square, so this valley reaches its floor at (0, −1).

  2. Go two units either side

    22−1=3;(−2)2−1=32^2-1=3;\quad(-2)^2-1=3

    Both two times two and minus two times minus two make four. Taking away one gives the same height on each side.

  3. Check the whole shape

    (−2,3),(0,−1),(2,3)(-2,3),\quad(0,-1),\quad(2,3)

    The curve falls toward the floor, then rises away from it. These three points help you reject a curve with the wrong height.

One floor, two matching sides

The valley reaches height minus one at its middle. At two units left or right, its height is three.

What makes the rule quadratic?

A quadratic rule has x² as its highest power of x. The curve is a parabola: a shape with two matching sides and one turn. It can open up like a valley or down like an arch.

In y = 2x² − 1, double the square before taking away one. The sides rise faster. In y = −x² + 3, the minus sign turns the square part down. The highest point is (0, 3).

These examples put the turn at x = 0. Other quadratic rules can move it left or right. Calculate from the full rule rather than trusting a U shape alone.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Calculate a point

y=2x2+1,x=−2y=2x^2+1,\qquad x=-2

Where does this graph sit when x is minus two?

Show the answerHide the answer
2⋅(−2)2+1=92\cdot(-2)^2+1=9

Square −2 to get 4. Multiply that by 2, then add 1.

Keep subtraction outside the square

22−1=(2−1)2  ?2^2-1=(2-1)^2\;?

Work out the height from the actual rule. Why do the added brackets change the result?

Show the answerHide the answer
22−1=3≠(2−1)22^2-1=3\ne(2-1)^2

The rule squares x before taking away 1. The brackets make a new rule.

Choose an arch instead of a valley

y=−x2+3y=-x^2+3

Both meet the side axis at three. Which has the right shape and height at x = 2?

Graph A

Its lowest point is (0, 3). At two, the height is seven.

Graph B

Its highest point is (0, 3). At two, the height is minus one.

Show the answerHide the answer
Graph B\text{Graph B}

The minus sign makes the curve open down. At x = 2, the rule gives minus one. Graph B fits both clues.

Test the turn and one more point

Match square rules to their graphs. Check whether the curve opens up or down, where it turns and one calculated point.

2-minute practice · keep going after mistakes · opens in a new tab

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