Question

Graph the function: y=(x−1)2+3
.

How is the graph a translation of f(x)=x2
?

(1 point)
Responses

Answer 1

To graph the function \( y = (x - 1)^2 + 3 \), we can analyze it in relation to the parent function \( f(x) = x^2 \):

1. **Basic Properties of the Function**:
- The parent function \( f(x) = x^2 \) is a parabola that opens upwards with its vertex at the origin (0,0).
- The function \( y = (x - 1)^2 + 3 \) represents a transformation of this parabola.

2. **Identify the Transformation**:
- The expression \( (x - 1) \) indicates a horizontal shift. Specifically, it shifts the graph to the right by 1 unit.
- The \( +3 \) indicates a vertical shift, moving the graph up by 3 units.

3. **Finding the Vertex**:
- From the form \( y = (x - 1)^2 + 3 \), the vertex of the parabola is at the point \( (1, 3) \).

4. **Graphing Steps**:
- Start by plotting the vertex at (1, 3).
- Since the parent function opens upwards, the transformed function will also open upwards.
- You can find additional points by selecting x-values around the vertex and calculating the corresponding y-values:
- For \( x = 0 \): \( y = (0 - 1)^2 + 3 = 1 + 3 = 4 \) → point (0, 4)
- For \( x = 2 \): \( y = (2 - 1)^2 + 3 = 1 + 3 = 4 \) → point (2, 4)
- For \( x = -1 \): \( y = (-1 - 1)^2 + 3 = 4 + 3 = 7 \) → point (-1, 7)
- For \( x = 3 \): \( y = (3 - 1)^2 + 3 = 4 + 3 = 7 \) → point (3, 7)

5. **Sketching the Graph**:
- Draw the vertex (1, 3).
- Draw the parabola opening upwards, passing through points (0, 4), (2, 4), (-1, 7), and (3, 7).

### Summary of Translation
- The graph of \( y = (x - 1)^2 + 3 \) is a translation of the function \( f(x) = x^2 \):
- Translated **1 unit to the right** (due to \( (x - 1) \)) and **3 units up** (due to \( +3 \)).

This makes the vertex of the graph on the point (1, 3) instead of (0, 0) for the parent function.

### Conclusion
The response concludes that the transformation involves a rightward shift by 1 unit and an upward shift by 3 units from the parent function \( f(x) = x^2 \).