Algebra, Year 9: curves, sequences and real-life graphs

1; −3; 6

HintSquare first, remembering what a negative times a negative gives, then do the subtraction.

The why(−2)² = 4, so y = 4 − 3 = 1; 0² − 3 = −3; and 3² − 3 = 6. The squares of 2 and −2 are equal, which is why the two halves of the curve mirror each other.

A parabola

HintA ball thrown through the air follows a path of this shape, turned upside down.

The whyEvery graph of the form y = ax² + bx + c is one of these. It has one line of symmetry, and it is curved all the way round: no straight sections and no sharp point at the bottom.

The graph is a smooth curve all the way along

HintTry x = 0.5: is its height halfway from the height at 0 to the height at 1?

The whyWhen x = 0.5, y = 0.25, but a ruler line from (0, 0) to (1, 1) would pass through 0.5. The true graph dips below each straight segment, so only a smooth freehand curve shows it correctly.

The object is not moving

HintTime is passing, but look at what is happening to the other quantity.

The whyAlong a flat section the time increases while the distance from the start stays the same, so nothing is travelling anywhere. A bus waiting at a stop draws a flat line.

She walks quickly, stops, then walks more slowly

HintSteepness shows speed: compare the three slopes.

The whySteep means a lot of distance in little time, so she is quick at first. Flat means no distance gained, so she has stopped. The gentler slope at the end covers distance less quickly.

It rises steeply at first, then more and more gently

HintThe same amount of water each second has to spread over a wider and wider surface.

The whyIn the narrow part a little water raises the level a lot, so the depth climbs quickly. As the vase widens, the same flow raises the level less each second, and the graph bends over and flattens.

speed

HintIt is measured in units such as metres per second.

The whyGradient is change in distance ÷ change in time, which is exactly how this quantity is defined. That is why a steeper line means a faster journey and a flat line means standing still.

10

HintUse only the last section: how far the level drops, and how long that takes.

The whyThe level drops 40 cm in 4 minutes, which is 10 cm every minute. It filled at only 40 ÷ 8 = 5 cm per minute, so on the graph the emptying section is twice as steep as the filling section.

A geometric sequence

HintIts name is shared with the branch of maths that deals with shapes.

The why3, 6, 12, 24 is one, because every term is double the one before. An arithmetic sequence adds the same amount each time; this kind multiplies by the same amount.

ratio

HintThe same word is used for comparisons such as 2 : 3.

The whyIn 3, 6, 12, 24 it is 2. The matching word for an arithmetic sequence is the common difference, the amount added each time.

5

HintDivide any term by the one just before it.

The why10 ÷ 2 = 5, 50 ÷ 10 = 5 and 250 ÷ 50 = 5. The multiplier between neighbours is the same all the way along, which is what makes the sequence geometric.

2

HintFind what each term is multiplied by; it does not have to be a whole number.

The whyEach term is one third of the one before (18 ÷ 54 = 1/3), so the next is 6 × 1/3 = 2. A common ratio between 0 and 1 makes a geometric sequence shrink towards zero without ever reaching it.

Sequence A: 2048 against 1200

HintCarry on the doubling for a few more terms than feels necessary.

The whyAt the 11th term A is still behind (1024 against 1100), but one more doubling takes it past. Adding gains the same amount every time; doubling gains more and more, so it overtakes any adding rule in the end.