The amplitude is half the distance between the highest point of the curve and the lowest point.

The crest is at \(y = 1\) and the trough is at \(y = -5\), so

\((1 - (-5)) \div 2 = 3\)

Amplitude \(= 3\)

It is a distance, so it is never negative — and it is measured from the curve's own centre line at \(y = -2\), not from the \(x\)-axis. Reading the crest and writing \(1\) is the slip this question is set to catch.

The period is how far along the \(x\)-axis the curve travels before it starts repeating itself.

Count the complete cycles between \(x = 0\) and \(x = 360\). There are \(1\) of them, so each one takes

\(360 \div 1 = 360\) degrees

Checking it the other way round: the distance from any crest to the next crest is \(360\) along the \(x\)-axis.

\(\sin x\) is at its largest when \(x = 90\), where it takes the value \(1\).

At that point \(y = 3 \times 1 - 2 = 1\)

So the maximum point is \((90, 1)\), which is the crest you can read straight off the graph.

\(\sin x\) is at its smallest when \(x = 270\), where it takes the value \(-1\).

At that point \(y = 3 \times (-1) - 2 = -5\)

So the minimum point is \((270, -5)\).

The crest and the trough sit the same distance either side of the line \(y = -2\), which is the curve's centre line.

\(\sin 180 = 0\), so at \(x = 180\) the curve is level with its own centre line.

\(y = 3 \times 0 - 2 = -2\)

The \(y\)-axis is the line \(x = 0\), so read the height of the curve there.

\(2^0 = 1\), so \(y = 6 \times 1 = 6\)

The curve crosses at \((0, 6)\).

Every graph of this shape crosses at the number in front of the power, because anything to the power \(0\) is \(1\).

Go up from \(x = 1\) to the curve, then straight across to the \(y\)-axis.

\(y = 6 \times 2^1 = 12\)

This time start on the \(y\)-axis at \(24\), go across to the curve, then straight down.

\(6 \times 2^x = 24\) means \(2^x = 24 \div 6\), and reading down from the curve gives

\(x = 2\)

Read the curve from left to right and see which way it goes.

This one shows exponential growth.

The give-away is the shape rather than the steepness. A growth curve climbs faster and faster; a decay curve falls quickly at first and then flattens out towards the horizontal axis without ever reaching it.

\(t = 0\) is where the curve meets the vertical axis.

Reading across from there gives \(10\).

Go up from \(t = 3\) to the curve, then straight across to the vertical axis.

The reading is \(80\).

Check it against part (b). The value is multiplied by the same amount at every step along the horizontal axis, so from \(10\) at \(t = 0\) it reaches \(80\) after \(3\) steps.

An angle in this range lies in the third quadrant, so the graphs (or CAST) give:

\(\sin y\) is negative, \(\cos y\) is negative and \(\tan y\) is positive.

Only box A pairs the signs of \(\mathrm{ cos }\,y\) and \(\mathrm{ tan }\,y\) this way.

Try successive powers of \(2\) until they pass \(48\):

\(2^{ 5 } = 32\), which is less than \(48\)

\(2^{ 6 } = 64\), which is more than \(48\)

So \(2^{ 5 } < 48 < 2^{ 6 }\), and \(x\) lies between \(5\) and \(6\).

The wave gives \(4\sin x + 1\) at every \(x\), and the flat line gives \(3\) at every \(x\). The two are equal exactly where the graphs cross.

They cross twice. Reading straight down from each crossing to the \(x\)-axis gives

\(x = 30\) and \(x = 150\)

Both answers matter. A wave is not a straight line, so the same height is reached twice — once on the way up and once on the way down.

The curve repeats every \(360\) degrees, so beyond the right-hand edge of the graph the picture starts again. Every solution therefore has another one \(360\) further along.

\(30 + 360 = 390\)

Adding \(360\) to the other solution would have been just as good; the question asks for one of them.

A tangent graph is drawn in separate branches. Look for the two places where the curve climbs away without limit on one side and comes back from far below on the other, with nothing joining the two.

\(x = 90\) and \(x = 270\)

These are the asymptotes. There is no point of the curve at either one, however far up or down you look, which is why the graph is drawn in pieces rather than as one line.

A tangent graph repeats from one asymptote to the next, rather than over a whole \(360\) the way a sine or cosine does.

The asymptotes here are at \(90\) and \(270\), so the period is

\(270 - 90 = 180\) degrees

\(\tan x\) is \(0\) at \(x = 180\), halfway between the two asymptotes.

\(y = 1 \times 0 - 2 = -2\)

Test the sketch against the three facts about \(y = 0.5^x\):

- it passes through \((0, 1)\), because anything to the power \(0\) is \(1\) - it is decreasing, because the base is less than \(1\) - it is a curve that never touches or crosses the \(x\)-axis, because a positive number to any power is positive

The sketch is a straight line, but the graph of an exponential function is a curve: its gradient changes all the way along.

One criticism of the sketch: The graph should be a curve, not a straight line.

Any number to the power \(0\) is \(1\), so \(y = 3^x\) passes through \((0, 1)\), the same point as \(y = 2^x\).

At \(x = 1\) the given curve is at \(y = 2\) and the new curve is at \(y = 3\). Since \(3\) is larger than \(2\), the new curve is above the given curve for \(x > 0\).

At \(x = -1\) the given curve is at \(y = \frac{ 1 }{ 2 }\) and the new curve is at \(y = \frac{ 1 }{ 3 }\). The new curve is below the given curve for \(x < 0\), closer to the \(x\)-axis.

Sketch a curve through \((0, 1)\), \((1, 3)\) and \((-1, \frac{ 1 }{ 3 })\), steeper than the given curve to the right of the \(y\)-axis and never reaching the \(x\)-axis on the left.