What are the important points to graph f(x)=sin(x)-5?

Dec 6, 2015

Shift the graph of $\sin \left(x\right)$ downwards of $5$ units.

Explanation:

Anytime you change a function from $f \left(x\right)$ to $f \left(x\right) + k$, you are making a vertical shift. In fact, in the old function you associated with every $x$ the $y$-value $f \left(x\right)$. Now, you're associating with every $x$ the new value $f \left(x\right) + k$, which is the old value with an additional constant. This means that you're changing the $y$ value, from the old ${y}_{0} = f \left(x\right)$ to the new ${y}_{1} = f \left(x\right) + k$. And as you can see, ${y}_{1} = {y}_{0} + k$.

So, if $k$ is positive, you're making the new $y$ bigger than the older, which means that the point $\left(x , {y}_{1}\right)$ is above the old point $\left(x , {y}_{0}\right)$. Otherwise, if $k$ is negative, the new point is below the old one.

So, in this case, you have the old function $y = \sin \left(x\right)$ that associates with every $x$ the value $\sin \left(x\right)$, and you're changing it with the new function $y = \sin \left(x\right) + 5$.

This new function works exactly like the old one, but it adds five extra units to the old value, which means that the graph is shifted upwards.