Trigonometric Graphs

Lesson

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`

From the lessons on key features and transformations, we discovered that

$a$a |
amplitude: the maximum deviation of the graph from the central line |

$\frac{2\pi}{b}$2πb |
period: the length needed for the curve to travel one full cycle |

$\frac{c}{b}$cb |
phase shift: the horizontal translation of the curve |

$d$d |
vertical translation: the shift of the central line (and hence the entire curve) up or down. |

In order to graph sine curves, there are a number of different approaches you can take. Which one you choose may depend on your own 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 **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

**Period **- the period of the function has been changed from $2\pi$2π to $\frac{2\pi}{\frac{1}{2}}=4\pi$2π12=4π so 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.

Step out all 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

$a$a |
amplitude | $2$2 |

sign of $a$a |
reflection | no reflection as $a$a is positive |

$\frac{2\pi}{b}$2πb |
period | $\frac{2\pi}{\frac{1}{2}}=4\pi$2π12=4π |

$\frac{c}{b}$cb |
phase shift | $0$0 |

$d$d |
vertical translation | $3$3 |

Step 2 - start by sketching 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 starting and ending position of the cycle (on the central line), also mark out the maximum and minimum points (on the quarter lines we sketched earlier).

Step 8 - sketch the curve lightly, joining our preparatory dots together. 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 they will all need to use the specific features of the sine curve.

Which of the following is the graph of $y=\sin x+4$`y`=`s``i``n``x`+4?

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Consider the function $y=\sin\left(x-\frac{\pi}{2}\right)$`y`=`s``i``n`(`x`−π2).

Identify the amplitude of the function.

Identify the phase shift of the function in radians.

Use a positive value to represent a shift to the right, and a negative value to represent a shift to the left.

Graph the function.

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Consider the function $y=3\sin\left(x-\frac{\pi}{3}\right)+2$`y`=3`s``i``n`(`x`−π3)+2.

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

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 and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions