Trigonometric Graphs

New Zealand

Level 7 - NCEA Level 2

Lesson

Previously we looked at key features and transformations of sine and cosine functions. We use these key concepts now to discuss graphing.

The general form of the equation of a sine curve is

$f\left(x\right)=a\sin\left(bx-c\right)+d$`f`(`x`)=`a``s``i``n`(`b``x`−`c`)+`d`

Recall that:

- The amplitude of the wave shape is given by the constant $a$
`a` - The period is determined from the coefficient $b$
`b`by dividing $360^\circ$360° (or $2\pi$2π radians) by $b$`b` - After writing the function as $f\left(x\right)=a\sin\left(b(x-\frac{c}{b})\right)+d$
`f`(`x`)=`a``s``i``n`(`b`(`x`−`c``b`))+`d`we see that the phase shift is $\frac{c}{b}$`c``b` - The vertical shift of the central line (and hence the entire curve) is given by the constant $d$
`d`.

To sketch the graphs of sine curves, a number of approaches can be taken. Which one you choose may depend on your preference or the question you are given.

Here are two approaches.

Walk through the transformations and change the stem graph $y=\sin x$`y`=`s``i``n``x` accordingly.

Graph $y=2\sin\left(\frac{x}{2}\right)+3$`y`=2`s``i``n`(`x`2)+3

Start with a sketch of $y=\sin x$`y`=`s``i``n``x`

Apply the vertical translation - move the graph up $3$3 units, this in now the graph of $y=\sin x+3$`y`=`s``i``n``x`+3

Increase the amplitude - the amplitude of this graph is $2$2 units, so we dilate the graph. Move the maximum and minimum out an extra unit. This is now the graph of $y=2\sin x+3$`y`=2`s``i``n``x`+3

The period of the function has been changed from $360^\circ$360° to $\frac{360^\circ}{\frac{1}{2}}=720^\circ$360°12=720°.This is a horizontal dilation. Stretch out the graph, keeping the starting point the same. This is now the graph of $y=2\sin\left(\frac{x}{2}\right)+3$`y`=2`s``i``n`(`x`2)+3. (Because the scale has been kept the same, only a half period is shown in the diagram.)

Step through the important components and create a dot-to-dot style map of the function.

Graph $y=2\sin\left(\frac{x}{2}\right)+3$`y`=2`s``i``n`(`x`2)+3

Step 1 - identify the transformations from the graph by identifying the following

- amplitude: $a=2$
`a`=2 - reflection: no reflection as $a$
`a`is positive - period: $\frac{360^\circ}{b}=\frac{360^\circ}{\frac{1}{2}}=720^\circ$360°
`b`=360°12=720° - phase shift: $\frac{c}{b}=0$
`c``b`=0 as $c=0$`c`=0 in this case - vertical translation: $d=3$
`d`=3

Step 2 - draw the central line (indicated by the vertical translation)

Step 3 - mark on the maximum and minimum by measuring the amplitude above and below the central line

Step 4 - check for a phase shift (this would shift the initial starting position)

There is no phase shift for this function as $c=0$`c`=0

Step 5 - mark out the distance of the full period. At this stage mark out half way and quarter way marks, this will help us sketch the curve.

Step 6 - check for a reflection (this would change the initial starting direction

There is no reflection for this function as $a>0$`a`>0

Step 7 - Create some dots on the starting and ending positions of the cycle (on the central line), also mark out the maximum and minimum points (on the quarter lines).

Step 8 - sketch the curve lightly, joining the preparatory dots. Developing the skills for smooth curve drawing takes practice so don't get disheartened.

Some people prefer step-by-step constructions, some prefer the fluid changes of transformations, others develop their own order and approach to sketching sine functions. Regardless of your approach, it will need to use the specific features of the sine curve.

Consider the function $y=\sin x-2$`y`=`s``i``n``x`−2.

Determine the period of the function, giving your answer in degrees.

Determine the amplitude of the function.

Determine the maximum value of the function.

Determine the minimum value of the function.

Graph the function.

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Consider the function $y=\sin\left(\frac{2}{3}x\right)$`y`=`s``i``n`(23`x`).

Identify the amplitude of the function.

Identify the period of the function, giving your answer in degrees.

Graph the function.

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Consider the function $y=3\sin2x-2$`y`=3`s``i``n`2`x`−2.

Determine the period of the function, giving your answer in degrees.

Determine the amplitude of the function.

Determine the maximum value of the function.

Determine the minimum value of the function.

Graph the function.

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Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems