To solve this problem, let's break it down into parts:
1. Identifying the Translation:
- We need to determine how many units and in which direction the function [tex]\( f(x) = 2x + 6 \)[/tex] is translated to match the graph of [tex]\( g \)[/tex].
2. Translation in the X-Direction:
- A horizontal translation by [tex]\( h \)[/tex] units to the right can be represented by [tex]\( f(x-h) \)[/tex].
- A horizontal translation by [tex]\( h \)[/tex] units to the left can be represented by [tex]\( f(x+h) \)[/tex].
3. Translation in the Y-Direction:
- A vertical translation by [tex]\( k \)[/tex] units up can be represented by [tex]\( f(x) + k \)[/tex].
- A vertical translation by [tex]\( k \)[/tex] units down can be represented by [tex]\( f(x) - k \)[/tex].
4. Determining the Function [tex]\( g(x) \)[/tex]:
- If the graph of [tex]\( g \)[/tex] is the graph of [tex]\( f \)[/tex] translated [tex]\(\square\)[/tex] units [tex]\(\square\)[/tex], the function [tex]\( g(x) \)[/tex] can be written in the form of the mentioned translations above.
Now let's fill in the blanks based on the options given and the steps we have:
- First, we need to choose the unit of translation (horizontal or vertical) and the direction (right, left, up, down).
- We then choose the respective function notation [tex]\( g(x) = \)[/tex] from the available options in the last set of choices.
Based on the problem given:
- The "units" should be a number (we need the specific number from the graph).
- The "direction" could be right, left, up, or down.
- The "last blank" should be filled with the appropriate function form from the given options that match our identified translation.
Inserting the appropriate selections from the options using the information determined above, we get:
The correct completion of the sentences from the provided options should be according to the details found in the graph translation.
Let's assume the graph translations (either horizontal or vertical and the number of units) fit:
If it is vertical and moves down:
1. "units" will be 5.
2. "direction" will be down.
3. The third blank, [tex]\( g(x) \)[/tex]= [tex]\( f(x)-5 \)[/tex].
This leads to:
"The graph of [tex]\( g \)[/tex] is the graph of [tex]\( f \)[/tex] translated 5 units down, and [tex]\( g(x) = f(x) - 5 \)[/tex]."
But, if it is horizontal right/left or any other number, adjust accordingly.
Be precise by referring directly to the actual graphical translation shifts!
