Hello!  Please could you explain: If we start with a function of y = x^2 , then we know that  y = x^2 + 2 is a y-shift upwards and y = (x + 2)^2 is an x-shift backwards.  However when we add extra x terms, the graph seems to be shifted in unpredictable ways both in y and x directions.  So for example, if we take :  y = x^2 + 6x  , the origin of (0,0) appears shifted to (-3,-9)  Is it possible to work out the x and y shift without working out either the roots or using calculus?

Yes, it is possible to determine the shift in the graph of the function \( y = x^2 + 6x \) without finding the roots or using calculus. The key is to rewrite the quadratic expression in a form that makes the shift clear, specifically by completing the square.

### Step-by-Step Process:

1. **Start with the original function:**
   \[
   y = x^2 + 6x
   \]

2. **Complete the square:**
   To rewrite the quadratic in a form that reveals the vertex (which indicates the shift), we complete the square. This involves manipulating the quadratic expression \( x^2 + 6x \) so it can be written as a perfect square plus a constant.

   - First, take the coefficient of \( x \), which is 6, divide it by 2, and then square the result:
     \[
     \left(\frac{6}{2}\right)^2 = 3^2 = 9
     \]
   - Add and subtract this square (9) within the expression to complete the square:
     \[
     y = x^2 + 6x = (x^2 + 6x + 9) - 9 = (x + 3)^2 - 9
     \]

3. **Rewrite the equation:**
   Now, the function is rewritten as:
   \[
   y = (x + 3)^2 - 9
   \]

   This form, \( y = (x + 3)^2 - 9 \), clearly shows the transformation of the original function \( y = x^2 \):

   - **Horizontal Shift:** The term \( (x + 3) \) indicates a shift to the left by 3 units. This means the graph is shifted from \( x = 0 \) to \( x = -3 \).
   - **Vertical Shift:** The term \(-9\) indicates a downward shift by 9 units. This means the graph is shifted from \( y = 0 \) to \( y = -9 \).

### Interpretation:
The vertex of the original parabola \( y = x^2 \), which is at \( (0,0) \), is shifted to the new vertex at \( (-3, -9) \).

### Conclusion:
Without using roots or calculus, we have determined that the function \( y = x^2 + 6x \) represents a parabola that has been shifted 3 units to the left and 9 units downward. The new vertex is at \( (-3, -9) \).

### Summary:
To determine the shift in a quadratic function \( y = ax^2 + bx + c \), completing the square is an effective method. It allows you to express the function in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex, revealing the shifts in the graph.

Would you like to explore further, or have any questions? Here are some related questions:

1. How would you complete the square for a quadratic function with a coefficient other than 1 in front of \( x^2 \)?
2. What are the effects of changing the coefficient of \( x^2 \) on the graph of a parabola?
3. Can you determine the vertex of a quadratic function using the formula without completing the square?
4. How do different forms of a quadratic equation (standard, vertex, factored) help in understanding the graph?
5. What is the geometric meaning of completing the square?

**Tip:** Completing the square is not only useful for finding the vertex of a parabola, but it also helps in solving quadratic equations and integrating certain functions.

Explore how to determine shifts in quadratic functions without finding roots or using calculus. Learn the step-by-step process of completing the square to identify horizontal and vertical shifts in parabolas. Suitable for high school students studying algebra.

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