Reading Trigonometric Graphs | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Read the midline, height and repeat

When the equation is missing, use features you can measure. The midpoint of the highest and lowest outputs gives the midline; half their difference gives the amplitude.

Solid curve: unknown function. Dashed line: y = 2.

Worked example

Find an equation for the solid curve

c=\frac{5+(-1)}2=2

|a|=\frac{5-(-1)}2=3

Consecutive maxima are at $x=0$ and $x=\pi$ , so the period is $\pi$ . Since $2\pi/b=\pi$ , choose $b=2$ .

It starts at a maximum, so positive cosine needs no shift:

Answer

y=3\cos(2x)+2

Check another point: at $x=\pi/2$ , cosine is −1 and the output is −1, matching the minimum.

An intersection is a solution

The dashed line is $y=2$ . Each intersection has the same $x$ and $y$ on both graphs, so it solves:

3\cos(2x)+2=2

There are four intersections on $0\le x\le2\pi$ , hence four solutions. When asked only for a number of solutions, count the intersections; a full algebraic solution is unnecessary. Include a touching point once and count an endpoint only when the interval includes it.

For tangent, use the asymptotes

The central zero is −π/4; adjacent asymptotes are −3π/4 and π/4.

This curve is tangent shifted left by $\pi/4$ . The original zero at 0 moves to $-\pi/4$ , and the original asymptotes at $\pm\pi/2$ move to $-3\pi/4$ and $\pi/4$ .

y=\tan(x+\pi/4)

The distance between adjacent asymptotes gives tangent's period. Do not use a sine amplitude rule: tangent has no highest or lowest output.

Your turn

Your turnPractice

A curve starts at a maximum of 7 when $x=0$ . Its minimum is 1 and its next maximum is at 120°. Find an equation using cosine. How many times does it meet $y=4$ on $0^\circ\le x\le360^\circ$ ?

Show worked answer

Midline $(7+1)/2=4$ ; amplitude $(7-1)/2=3$ ; $360^\circ/b=120^\circ$ gives $b=3$ .

Answer

y=3\cos(3x)+4

There are three full cycles, each crossing the midline twice: six intersections. Neither endpoint lies on the midline.

Read the midline, height and repeat

When the equation is missing, use features you can measure. The midpoint of the highest and lowest outputs gives the midline; half their difference gives the amplitude.

Solid curve: unknown function. Dashed line: y = 2.

Worked example

Find an equation for the solid curve

c=\frac{5+(-1)}2=2

|a|=\frac{5-(-1)}2=3

Consecutive maxima are at $x=0$ and $x=\pi$ , so the period is $\pi$ . Since $2\pi/b=\pi$ , choose $b=2$ .

It starts at a maximum, so positive cosine needs no shift:

Answer

y=3\cos(2x)+2

Check another point: at $x=\pi/2$ , cosine is −1 and the output is −1, matching the minimum.

An intersection is a solution

The dashed line is $y=2$ . Each intersection has the same $x$ and $y$ on both graphs, so it solves:

3\cos(2x)+2=2

For tangent, use the asymptotes

The central zero is −π/4; adjacent asymptotes are −3π/4 and π/4.

This curve is tangent shifted left by $\pi/4$ . The original zero at 0 moves to $-\pi/4$ , and the original asymptotes at $\pm\pi/2$ move to $-3\pi/4$ and $\pi/4$ .

y=\tan(x+\pi/4)

The distance between adjacent asymptotes gives tangent's period. Do not use a sine amplitude rule: tangent has no highest or lowest output.

Your turn

Your turnPractice

A curve starts at a maximum of 7 when $x=0$ . Its minimum is 1 and its next maximum is at 120°. Find an equation using cosine. How many times does it meet $y=4$ on $0^\circ\le x\le360^\circ$ ?

Show worked answer

Midline $(7+1)/2=4$ ; amplitude $(7-1)/2=3$ ; $360^\circ/b=120^\circ$ gives $b=3$ .

Answer

y=3\cos(3x)+4

There are three full cycles, each crossing the midline twice: six intersections. Neither endpoint lies on the midline.