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. |
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.
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?
Which of the following is the graph of $y=\sin x+4$y=sinx+4?
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.
The function $y=\sin x$y=sinx is translated $4$4 units up.
Determine the equation of the new function after the translation.