To identify the transformation from [tex]\( f(x) = x^2 - 1 \)[/tex] to [tex]\( g(x) = (x + 2)^2 - 1 \)[/tex], let's analyze the changes step-by-step.
1. Start with the original function: [tex]\( f(x) = x^2 - 1 \)[/tex].
2. Consider the function [tex]\( g(x) = (x + 2)^2 - 1 \)[/tex].
First, recognize that the main difference between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is the argument of the squared term, [tex]\( x \)[/tex] vs. [tex]\( x + 2 \)[/tex].
### Step-by-Step Analysis:
#### Step 1: Compare the squared terms
- In [tex]\( f(x) \)[/tex], the squared term is [tex]\( x^2 \)[/tex].
- In [tex]\( g(x) \)[/tex], the squared term is [tex]\( (x + 2)^2 \)[/tex].
#### Step 2: Understand the transformation
- The term [tex]\( x + 2 \)[/tex] indicates a horizontal translation because it involves adding a constant to the [tex]\( x \)[/tex]-variable inside the function.
- Specifically, adding 2 to [tex]\( x \)[/tex] suggests shifting the graph of the function to the left.
Therefore, [tex]\( g(x) = (x + 2)^2 - 1 \)[/tex] is the same as shifting the graph of [tex]\( f(x) = x^2 - 1 \)[/tex] horizontally.
#### Step 3: Determine the direction and magnitude of the shift
- Adding a constant [tex]\( +2 \)[/tex] to [tex]\( x \)[/tex] shifts the graph to the left by 2 units.
Thus, the correct description of the transformation from [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex] is:
a horizontal translation 2 units to the left.
So, the answer is:
- a horizontal translation 2 units to the left.
This analysis confirms that this transformation is correctly described as a horizontal translation of 2 units to the left.
