Question

On a graph there is an upward v shape with points (-3,1),(-2,2),(-3,1),(-4,0),(-5,1),(-6,2),(-7,3)
Interpret the key features and type of transformation
Absolute value function has undergone a vertical translation
An absolute value function has undergone a horizontal translation
A quadratic function has undergone a vertical translation
A quadratic function has undergone a horizontal translation

Answer 1

To analyze the given points and determine the type of function and its transformations, let's first identify the shape of the graph defined by these points:

The points are:
- (-3,1)
- (-2,2)
- (-4,0)
- (-5,1)
- (-6,2)
- (-7,3)

Plotting these points, we can see the upward and downward trends, suggesting the resemblance of an "upward V" shape, characteristic of an absolute value function.

### Characteristics of the Function:
1. **V-Shape**: The points (-3,1), (-2,2), and (-4,0) indicate that as you move away from the central point (-3,1), the function behaves symmetrically. This matches the behavior of the absolute value function which has a vertex, around which the points form a V shape.

2. **Vertex Location**: The vertex is at (-3, 1), which suggests that it has been translated vertically and horizontally.

### Transformation Analysis:
The standard absolute value function can be described by:
\[ f(x) = |x| \]
The general form for an absolute value function with transformations is:
\[ f(x) = a |x - h| + k \]
where (h, k) is the vertex of the "V".

### Identifying Vertical and Horizontal Translations:
From our identified vertex:
- **Horizontal Translation**: The graph has moved from the original position at the vertex of (0, 0) to (-3, 1). This shows that the graph has undergone a **horizontal translation to the left by 3 units**.

- **Vertical Translation**: The graph has also been translated up by 1 unit, as evident from the change in the y-coordinate.

### Conclusion:
Since the function matches the form of an absolute value function and has undergone both vertical and horizontal translations, the correct interpretation of the key features is that:

**An absolute value function has undergone both a vertical translation (upward by 1 unit) and a horizontal translation (to the left by 3 units).**

If you have to choose only one of the options given, the best fit would be:
- **An absolute value function has undergone a vertical translation** (since this emphasizes the vertical movement, but keep in mind that the horizontal translation is also significant).