Start from the parent function f(x) = sqrt(x), and think about how each of the given graphs relates to it.
B
b
Look at the given functions. Can you rewrite one of them as a basic transformation of the other?
A
a
Graph of y = sqrt(- x):
Graph of y = sqrt(1 - x):
Graph of y = sqrt(2 - x):
B
b
See solution.
Practice makes perfect
a
Let's start by recalling the parent function f(x) = sqrt(x). We can see its graph in the figure below.
As we can see, this function is very similar to the ones given in the exercise. Let's label each of them differently to distinguish between them.
y_1 = sqrt(- x) y_2=sqrt(1-x) y_3=sqrt(2-x)Notice that y_1 is the most similar to the parent function. We can rewrite y_1 as a transformation of it.
y = sqrt(- x) = f(- x)
Now we can see that y_1 = sqrt(- x) is a reflection in the y-axis of the parent function. We can use this information to graph it.
To graph y_2=sqrt(1-x) and y_3 = sqrt(2-x), we can do something similar as to what we did above. Let's rewrite them as a transformation of y_1, which we already know how to graph.
sqrt(1-x) ⇔ sqrt(- (x -1 ))=y_1(x-1)
sqrt(2-x) ⇔ sqrt(- (x -2 ))=y_1(x-2)
We can see that they are the same function, but y_2 is translated 1 unit to the right with respect to y_1, and y_2 is translated 2 units to the right. Let's graph y_2= sqrt(1-x).
Now we can graph y_3= sqrt(2-x).
b
Notice that they are very similar — in fact, the only difference is that they have the opposite signs in their radicands, which are the arguments of these functions.
sqrt(x-h) = sqrt(- (h-x))
Therefore, one is the reflection in the y-axis of the other.
