Transformation: Reflection in the x-axis, a vertical stretch by a factor 2 and a vertical translation up by 3 units. Graph:
Practice makes perfect
We want to describe the transformations of the parent function f(x)=sqrt(x) represented by g(x)=sqrt(- 32x)+3. Let's start by rewriting g(x) into an easier form.
g(x)=sqrt(- 32x)+3
⇕
g(x)=-2sqrt(x)+3
Let's look at the possible transformations. Then we can more clearly identify the ones being applied.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Vertical Stretch or Shrink
Vertical stretch, a>1
y= af(x)
Vertical shrink, 0< a<1
y= af(x)
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
Now, using the table, let's highlight the transformations.
g(x)=- 2sqrt(x)+ 3
We can describe the transformations as a reflection in the x-axis, a vertical shrink by a factor 2 and a vertical translation up by 3 units. Let's begin by graphing the parent function.
Next, we will multiply the y- coordinates by -1. This reflects the graph across the x-axis.
Next, we will divide the x-coordinates by a= 2. This stretches the parent graph by a factor of 2.
Next, we will translate the graph 3 units up. To do so we will add 3 to every y-coordinate.
Finally, let's show the graphs of f(x) and g(x) on the same coordinate plane. Remember, that g(x) = - 2 sqrt(x) + 3 is equal to g(x)=sqrt(- 32x)+3.
