How do you graph y=-1/2sqrt(x-4)+1?

Sep 3, 2016

Start with the graph of $y = \sqrt{x}$. Remember that this means our domain is $x > 0$, and we can graph the points $\left(0 , 0\right) , \left(1 , 1\right) , \left(4 , 2\right) , \left(9 , 3\right) , \left(16 , 4\right)$, and so on.

graph{sqrtx [-4.73, 35.82, -8.76, 11.51]}

The next step is to graph $y = \sqrt{x - 4}$. Notice that at $x = 4$, this gives us $\sqrt{0} = 0$. At $x = 13$, we have $\sqrt{13 - 4} = \sqrt{9} = 3$. What adding the $- 4$ within the function does is actually shift the function $4$ units to the right. (Think about comparing the point on $y = \sqrt{x}$ of $\left(9 , 3\right)$ with the point on $y = \sqrt{x - 4}$ of $\left(13 , 3\right)$--that's the shift.)

graph{sqrt(x-4) [-4.73, 35.82, -8.76, 11.51]}

Next, we can graph $y = \frac{1}{2} \sqrt{x - 4}$. This means you can take whatever $y$ value previously existed, and halve it. So, the point at $\left(20 , 4\right)$ will become $\left(20 , 2\right)$ and the point at $\left(5 , 1\right)$ will become $\left(5 , \frac{1}{2}\right)$.

graph{1/2sqrt(x-4) [-4.73, 35.82, -8.76, 11.51]}

For $y = - \frac{1}{2} \sqrt{x - 4}$, just reflect all the points over the $x$-axis, which is the same as taking the negative versions of all the existing $y$-values.

graph{-1/2sqrt(x-4) [-4.73, 35.82, -8.76, 11.51]}

Finally, for $y = - \frac{1}{2} \sqrt{x - 4} + 1$, take the graph and shift it upwards $1$ point:

graph{-1/2sqrt(x-4)+1 [-4.73, 35.82, -8.76, 11.51]}