Perform and recognise vertical translation for sine and cosine curves and equations (rad)

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.

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=sinx−k, how might the value of $k$k change?

Practice questions

Question 1

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QUESTION 3