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The problem asks you to start with the parabola equation \( y = x^2 - 6x + 5 \), reflect it along the x-axis, and then shift it upward by 2 units. After these transformations, you're asked to find the value of the new parabola at \( x = 5 \).

Let's go step-by-step:

### 1. Reflect along the x-axis:
Reflecting a function \( y = f(x) \) across the x-axis means you multiply the entire function by \(-1\). So, the new equation after reflection becomes:
\[
y = -(x^2 - 6x + 5) = -x^2 + 6x - 5
\]

### 2. Shift upward by 2 units:
Shifting a function upward by 2 units means adding 2 to the function. So, the new equation after the shift is:
\[
y = -x^2 + 6x - 5 + 2 = -x^2 + 6x - 3
\]

### 3. Evaluate at \( x = 5 \):
Now, substitute \( x = 5 \) into the new equation:
\[
y = -(5)^2 + 6(5) - 3 = -25 + 30 - 3 = 2
\]

So, the value of the new parabola at \( x = 5 \) is \( y = 2 \).

Would you like a more detailed explanation of any steps?

Here are 5 related questions to expand your understanding:

1. How do horizontal reflections affect the graph of a parabola?
2. What happens if you shift a parabola downward instead of upward?
3. Can you derive the vertex form of the given parabola before applying transformations?
4. How would horizontal shifts affect this transformation?
5. What is the significance of completing the square in transforming quadratic equations?

**Tip:** When reflecting a function across the x-axis, always remember to multiply the whole equation by \(-1\), and apply any shifts afterwards.

Consider the parabola y = x^2 - 6x + 5. Suppose you reflect it along the x axis, and shift it up two units. Find the value of the new parabola at x = 5.

Learn how to solve a quadratic transformation problem. This involves reflecting the parabola y = x^2 - 6x + 5 along the x-axis, shifting it upward by 2 units, and finding the value at x = 5. Detailed step-by-step explanation included. Suitable for Grades 10-12.

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