Question
The equation \( y=x^{2} \) is graphed in the standard \( (x, y) \) coordinate plane below. In which of the following equations is the graph of the parabola shifted 4 units to the left and 2 units up?
Ask by Lyons Hall. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation after shifting is \( y = (x + 4)^2 + 2 \).
Solution
To shift the graph of the parabola \( y = x^{2} \) **4 units to the left** and **2 units up**, you'll need to adjust the equation accordingly.
### Step-by-Step Transformation:
1. **Shifting Left or Right:**
- **Left Shift:** To shift the graph to the **left** by \( h \) units, replace \( x \) with \( x + h \).
- **Right Shift:** To shift the graph to the **right** by \( h \) units, replace \( x \) with \( x - h \).
For a **4-unit left shift**:
\[
y = (x + 4)^{2}
\]
2. **Shifting Up or Down:**
- **Up Shift:** To shift the graph **up** by \( k \) units, add \( k \) to the entire equation.
- **Down Shift:** To shift the graph **down** by \( k \) units, subtract \( k \) from the entire equation.
For a **2-unit upward shift**:
\[
y = (x + 4)^{2} + 2
\]
### Final Equation:
After applying both transformations, the new equation of the parabola is:
\[
y = (x + 4)^{2} + 2
\]
This equation represents the original parabola \( y = x^{2} \) shifted **4 units to the left** and **2 units up** in the coordinate plane.
**Answer:** After shifting, the equation is y = (x + 4) squared plus two. In symbols,
y = (x+4)² + 2
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To shift the graph of the parabola \( y = x^2 \) 4 units to the left, you replace \( x \) with \( x + 4 \). To shift it 2 units up, you add 2 to the entire equation. Therefore, the new equation becomes: \[ y = (x + 4)^2 + 2 \] This means the new equation represents the same shape of the parabola but repositioned in the coordinate plane. You can visualize this by how points that originally were at (0,0) would now be at (-4,2), creating that desired shift. Keep in mind that while transforming equations might initially feel tricky, working through these shifts and recognizing how they affect the graph can be an exciting puzzle, especially when you see how they change the vertex's location!
