Answer:
The function is a quadratic function.
Series of transformations:
- Horizontal translation of 3 units left.
- Vertical stretch by a factor of 2.
- Reflection in the x-axis.
- Vertical translation of 1 unit up.
Explanation:
A quadratic function is a polynomial function of degree 2 (the highest exponent of the variable is 2).
Therefore, the given equation, y = -2(x + 3)² + 1, represents a quadratic function as the term (x + 3)² indicates that the highest power of the x-variable is 2.
The parent function of a quadratic function is y = x².
To translate the parent function to produce the given equation, we need to apply this series of transformations:
1. Horizontal translation
When "a" is added to the x-variable, the graph is shifted "a" units to the left. Therefore, the addition of 3 inside the parentheses shifts the graph horizontally to the left by 3 units.
2. Vertical Stretch
The coefficient in front of the squared term (x + 3)² indicates a vertical stretch or compression. In this case, since the coefficient greater than 1, it stretches the graph vertically, making it narrower compared to the parent function.
3. Reflection
As the stretch coefficient is negative, the graph is reflected across the x-axis, meaning the parabola opens downwards.
4. Vertical translation
When "a" is added to the function, the graph is shifted "a" units up. Therefore, the addition of 1 to the function shifts the graph vertically up by 1 unit.
In summary, the series of transformations that maps the parent function to the given function is:
- Horizontal translation of 3 units left.
- Vertical stretch by a factor of 2.
- Reflection in the x-axis.
- Vertical translation of 1 unit up.