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This page sets out the method for trigonometric & exponential graphs in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Trigonometric & exponential graphs topic page.
When your calculator (or exact values) gives one solution of or and the question wants all solutions in .
When an exponential graph passes through two known points, use the point with first.
On the sine graph, the value at 90 degrees is 1, at 180 degrees is 0, and at 270 degrees is -1.
On the cosine graph, the value at 0 degrees is 1 and at 180 degrees is -1.
On the tangent graph, the value at 45 degrees is 1.
The period of the sine graph in degrees is 360. The shape then repeats.
On the exponential graph, the value of 2³ is 8, and the value of 2⁰ is 1.
Sine starts at 0. Cosine starts at 1. The exponential graph does not wave.
Mark the five easy points before you sketch. For sine they are (0, 0), (90, 1), (180, 0), (270, -1) and (360, 0). The curve is smooth and it crosses the axis only at 0^°, 180^° and 360^° in this range. Cosine is the same wave shifted: (0, 1), (90, 0), (180, -1), (270, 0), (360, 1). If you can draw one, you can draw the other by moving it 90^° to the left.
Reading a value is the reverse of plotting a point. sin 30^°=1/2, so the horizontal line y=1/2 meets the sine curve twice between 0^° and 360^°: at 30^° and at 180^°-30^°=150^°. A calculator in degree mode gives you the first of these. The graph gives you the second. Quoting only 30^° is half an answer on almost every question of this type.
Method: 1. Sketch the five landmark points before you solve. 2. A calculator gives the first solution in degree mode. 3. Use the symmetry of the curve for the second solution. 4. Reject any angle outside the stated range.
Cosine is positive in the first and fourth quadrants. If cos x=1/2, the acute angle is 60^°, and the second solution is 360^°-60^°=300^°. Using 180^°-60^°=120^° is the sine rule of symmetry applied to the wrong function. cos 120^°=-1/2, not 1/2. Say the name of the function before you choose the symmetry.
Between 0^° and 360^° there are still two solutions for a cosine value that is not 1 or -1. At 1 the solutions meet at 0^° and 360^°, which some questions treat as the same direction. At 0 the solutions are 90^° and 270^°. Sketch a short horizontal line across your cosine curve and count the crossings. That count is the number of solutions you should write.
Method: 1. The second positive cosine is 360^° minus the first angle. 2. Do not use the sine rule 180^°-x on a cosine question. 3. Check the sign of cosine in the quadrant you chose. 4. Count the crossings of the horizontal line to see how many answers you owe.
sin x=-1/2 is not solved by 180^°-30^°. That gives the positive value. The related acute angle is still 30^°, but the sine is negative in the third and fourth quadrants. The third-quadrant angle is 180^°+30^°=210^°. The fourth is 360^°-30^°=330^°. Check the signs: sine of an angle in those two blocks is negative, and the size of the related angle is 30^°.
Tangent repeats every 180^° and it is undefined at 90^° and 270^°. Draw dashed vertical lines there and do not join the curve across them. In each 180^° block, y=tan x takes every real value once. So tan x=1 has solutions 45^° and 45^°+180^°=225^° in the full turn. Adding 180^°, not using the sine or cosine symmetries, is the tangent method.
Method: 1. Find the related acute angle from the positive value. 2. Negative sine is in the third and fourth quadrants. 3. Tangent repeats every 180^° and is undefined at 90^° and 270^°. 4. Do not join a tangent curve across an asymptote.
y=3^x passes through (0, 1) because 3^0=1, through (1, 3), through (2, 9) and through (3, 27). To the left, 3^-1=1/3 and 3^-2=1/9. The heights get smaller but they stay positive, so the curve approaches the x-axis and does not cross it. It does not pass through the origin. A sketch that cuts the x-axis is a different kind of graph.
If y=ab^x and you know two points, find a from the y-intercept and b from a second point. Through (0, 1) and (2, 9), a=1 and b^2=9. Since the curve is increasing, b=3 rather than b=-3. Then y=3^x, and at x=3 the height is 27. The same reading works for y=2^x: (0, 1) and (3, 8), because 2^3=8.
Method: 1. k^0=1, so the curve crosses the y-axis at (0,1). 2. For k>1 the curve rises and stays positive. 3. A negative power is a reciprocal, not a negative height. 4. The curve approaches the x-axis and does not cross it.
Exam questions on these graphs are often a sketch with one labelled point. For y=sin x from 0^° to 360^° the turning points are (90^°, 1) and (270^°, -1). For y=cos x they are (0^°, 1), (180^°, -1) and (360^°, 1). Learn the pattern of values 0, 1, 0, -1, 0 for sine and 1, 0, -1, 0, 1 for cosine at the multiples of 90^°.
When you are asked to sketch, the marks go for the shape and for the points where the curve crosses the axes, not for a perfect drawing. Label the x values you know, mark the maximum and minimum heights, and draw a smooth curve with no straight sections. For y=tan x draw dashed vertical lines at x=90^° and x=270^° and keep the curve on the correct side of them.
Method: 1. Write the five key x values 0, 90, 180, 270, 360 along the axis. 2. Write the matching y values from memory: sine gives 0, 1, 0, -1, 0, cosine gives 1, 0, -1, 0, 1. 3. Plot the five points and join them with a smooth curve. 4. For a tangent graph add the asymptotes at 90^° and 270^° before you draw the branches.
Tip: A turning point question wants both coordinates. Writing only the angle, or only the height, loses the mark.
Some questions give a graph for 0^° to 360^° and then ask for all solutions up to 720^°. The graph repeats every 360^°, so each solution in the first turn gives another one 360^° later. A request for four solutions of sin x=0.6 means two in the first turn and two in the second.
You may also be told one value, such as sin 53.1^°=0.8, and be asked to use it for a different equation. Use the symmetry of the curve: sine repeats as 180^°-x, and a negative value sits in the third and fourth quadrants at 180^°+x and 360^°-x. Write the sums down so the method mark is secure even if the arithmetic slips.
Method: 1. Find the first solution, or use the one you are given. 2. Use the symmetry of the curve to find the second solution in the same turn. 3. Add 360^° to each to reach the next turn. 4. Check that every answer sits inside the range in the question, and that the number of answers matches the question.
Transformation questions on trigonometric graphs follow the same rules as any other curve. Adding a number outside, as in y=cos x+2, moves the whole graph up by 2. Putting a number inside the bracket, as in y=cos(x-90^°), moves it right by 90^°, the opposite way to the sign. A minus sign outside, as in y=-sin x, reflects the graph in the x-axis.
The usual exam wording is to draw or describe the new graph, or to say where a labelled point moves to. Track one key point, such as the maximum, rather than the whole curve. For a tangent graph that has been translated, compare the old and new coordinates of one point to find the translation, then write it as an expression.
Method: 1. Pick one key point on the original graph, such as a maximum or a point on the axis. 2. Apply the shift: outside the bracket moves up or down, inside moves left or right the opposite way. 3. Write the new coordinates of the point and check they fit the new equation. 4. Draw or describe the full curve using the new key points.
An exponential relationship is often written y=a× b^x. When x=0 the power b^0 is 1, so the curve crosses the y-axis at (0, a). This is the quickest way to find a. If b>1 the graph grows, and if 0<b<1 it decays towards the x-axis without touching it. Say aloud what the letters mean before you start, because the constant a and the base b do different jobs.
To find b, put a second point into the equation and solve. If the curve passes through (0, 5) and (3, 40), then a=5 and 5b^3=40, so b^3=8 and b=2. In a context such as bacteria or savings, t replaces x and the first number a is the starting amount. Always say what the answer means with its units.
Method: 1. Put x=0 to read off a, the starting value. 2. Substitute the second point into y=a× b^x. 3. Divide by a, then take the root to find b. 4. Use the finished equation to answer the question and check with the second point.
Tip: Check the equation against both points before you use it. If either fails, find the slip first.
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