https://r2.erweima.ai/mathbot/image/1726475326282hznmarby.jpg

The problem you uploaded is asking how to translate the graph of the quadratic function \( y = x^2 - 8x - 13 \) so that it overlaps with the graph of another quadratic function, \( y = x^2 + 4x + 3 \).

### Step-by-step Solution:

1. **Identify the general form of the equations:**
   Both are quadratic functions in the form of \( y = ax^2 + bx + c \), where \( a = 1 \) in both cases, indicating that the graphs have the same shape (parabolas opening upwards), but their positions differ.

2. **Compare the two functions:**

   - First function: \( y_1 = x^2 - 8x - 13 \)
   - Second function: \( y_2 = x^2 + 4x + 3 \)

   We need to determine how to shift \( y_1 \) to match \( y_2 \).

3. **Rewrite the functions to compare translations:**

   - For \( y_1 = x^2 - 8x - 13 \), completing the square:
     \[
     y_1 = (x^2 - 8x) - 13 = (x - 4)^2 - 16 - 13 = (x - 4)^2 - 29
     \]
     The vertex of this parabola is at \( (4, -29) \).

   - For \( y_2 = x^2 + 4x + 3 \), completing the square:
     \[
     y_2 = (x^2 + 4x) + 3 = (x + 2)^2 - 4 + 3 = (x + 2)^2 - 1
     \]
     The vertex of this parabola is at \( (-2, -1) \).

4. **Find the horizontal and vertical shifts:**

   - To match \( y_1 \) with \( y_2 \), we need to shift the vertex of \( y_1 = (x - 4)^2 - 29 \) to \( (-2, -1) \).
   - The horizontal shift is from \( x = 4 \) to \( x = -2 \), so we need to shift 6 units to the left.
   - The vertical shift is from \( y = -29 \) to \( y = -1 \), so we need to shift 28 units up.

### Conclusion:
To make the graph of \( y = x^2 - 8x - 13 \) coincide with the graph of \( y = x^2 + 4x + 3 \), we need to shift the graph **6 units to the left** and **28 units up**.

Would you like more details on this solution, or do you have further questions?

### Related Questions:
1. How do you complete the square for a general quadratic function?
2. What is the geometric meaning of translating a quadratic function?
3. How does the axis of symmetry change when translating a parabola?
4. What happens to the vertex form of a quadratic equation during a translation?
5. Can you translate the graph of any quadratic function to match another without changing its shape?

### Tip:
When translating quadratic graphs, always focus on how the vertex moves since the shape of the parabola remains unchanged unless there is a reflection or a change in its "width."

2次関数 y = x^2 - 8x - 13 のグラフをどのように平行移動すると、2次関数 y = x^2 + 4x + 3 のグラフに重なるか。

Discover how to shift the graph of the quadratic function y = x^2 - 8x - 13 to overlap with y = x^2 + 4x + 3 using horizontal and vertical translations. This detailed guide includes completing the square and vertex form transformations, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation