Quiz
Sign In
Consider vertical and horizontal translations, stretches and shrinks, and reflections.
Transformations: The graph of h is a a horizontal translation 2 units left, a reflection in the x-axis, a vertical stretch by a factor of 3 and a vertical translation 6 units up of the graph of f.
Graph:
Let's first describe the transformations from the graph of f to the graph of h. Then, we will graph h.
We want to describe the transformations of the parent function f(x)=sqrt(x) represented by h(x)=- 3sqrt(x+2)+6. To do so, 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 | |
| Horizontal Translations | Translation right h units, h>0 y=f(x- h) |
| Translation left h units, h>0 y=f(x+ h) | |
| 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.
h(x)=- 3sqrt(x+ 2)+ 6
We can describe the transformations as a horizontal translation 2 units left, a reflection in the x-axis, a vertical stretch by a factor of 3 and a vertical translation up by 6 units.
Let's start by finding the domain of h(x)=- 3sqrt(x+2)+6. To do so, recall that the radicand of a square root is always greater than or equal to 0. x+2≥ 0 ⇔ x≥ -2 Therefore, the domain of the given function is all real numbers greater than or equal to -2. With this in mind, we will make a table of values to graph the function.
| x | - 3sqrt(x+2)+6 | h(x)=- 3sqrt(x+2)+6 |
|---|---|---|
| -2 | - 3sqrt(-2+2)+6 | 6 |
| -1 | - 3sqrt(-1+2)+6 | 3 |
| 2 | - 3sqrt(2+2)+6 | 0 |
| 7 | - 3sqrt(7+2)+6 | -3 |
Let's plot and connect the obtained points. Remember, the domain is all real numbers greater than or equal to - 2, so we do not want to extend the function any farther to the left.
