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Grade 12

Vertical translation for sine and cosine

Lesson

In general, we say that the graph of a function $y=f\left(x\right)$y=f(x) is vertically translated when the resulting graph is of the form $y=f\left(x\right)+c$y=f(x)+c, where $c$c is some non-zero constant.

Graphically speaking, a vertical translation takes each point on the graph of $y=f\left(x\right)$y=f(x) and adds (or subtracts) a constant to the $y$y-value of each point. For instance, we might have the graph of $y=\cos x$y=cosx as shown below, and all the points shift upwards according to the constant term.

$y=\cos x$y=cosx vertically translated upwards by a positive constant $c$c.

 

The constant term does not necessarily have to be positive. In the case that $c$c is negative, that is $c<0$c<0, the graph of a function will vertically translate downwards.

$y=\sin x$y=sinx vertically translated downwards by a negative constant $c$c.

 

Careful!

For a negative value of $c$c, the graph of $y=f\left(x\right)$y=f(x) translates vertically downwards, although we would still write the equation of the resulting graph as $y=f\left(x\right)+c$y=f(x)+c.

Alternatively, we might instead write $y=f\left(x\right)-c$y=f(x)c, where $c$c is redefined as the absolute value.

Worked example

The graph of $y=\sin x+k$y=sinx+k has been vertically translated upwards by $5$5 units from $y=\sin x$y=sinx. What is the value of $k$k?

Think: The equation $y=\sin x+k$y=sinx+k is of the form $y=f\left(x\right)+c$y=f(x)+c where $c$c determines the direction and the magnitude of the vertical translation.

Do: A positive value of $k$k will translate the graph of $y=\sin x$y=sinx upwards. So $k=5$k=5.

Reflect: If instead we asked about the graph of $y=\sin x-k$y=sinxk, how might the value of $k$k change?

Practice questions

Question 1

Which of the following is the graph of $y=\sin x+4$y=sinx+4?

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    A

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    B

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    C

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    D

QUESTION 2

A graph of $y=\cos x$y=cosx is given.

Plot the resulting graph when $y=\cos x$y=cosx is translated $4$4 units down.

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QUESTION 3

The function $y=\sin x$y=sinx is translated $4$4 units up.

  1. Determine the equation of the new function after the translation.

Outcomes

12CT.C.2.3

Determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = sin (x – d) + c and y = cos (x – d) + c, and describe these roles in terms of transformations on the graphs of f(x) = sin x and f(x) = cos x with angles expressed in degrees

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