Question

What happens to the accents intercepts of the function y equals x squared minus 25 when the graph is reflected over the x-axis

Answer 1

To analyze the effects of reflecting the function \( y = x^2 - 25 \) over the x-axis, let's first find the intercepts of the original function.

1. **Finding the x-intercepts**:
The x-intercepts occur where \( y = 0 \).
\[
0 = x^2 - 25
\]
This can be factored as:
\[
0 = (x - 5)(x + 5)
\]
Thus, the x-intercepts are:
\[
x = 5 \quad \text{and} \quad x = -5
\]
So, the x-intercepts are \( (5, 0) \) and \( (-5, 0) \).

2. **Finding the y-intercept**:
The y-intercept occurs where \( x = 0 \):
\[
y = 0^2 - 25 = -25
\]
Thus, the y-intercept is:
\[
(0, -25)
\]

Next, let's reflect the function over the x-axis. Reflecting a function \( y = f(x) \) over the x-axis changes the function to \( y = -f(x) \).

For our function:
\[
y = x^2 - 25 \quad \text{becomes} \quad y = -(x^2 - 25) = -x^2 + 25
\]

Now we can determine the intercepts of the reflected function \( y = -x^2 + 25 \).

1. **Finding the x-intercepts of the reflected function**:
Set \( y = 0 \):
\[
0 = -x^2 + 25
\]
Rearranging gives:
\[
x^2 = 25
\]
This leads to:
\[
x = 5 \quad \text{and} \quad x = -5
\]
Thus, the x-intercepts remain the same as the original function: \( (5, 0) \) and \( (-5, 0) \).

2. **Finding the y-intercept of the reflected function**:
Set \( x = 0 \):
\[
y = -0^2 + 25 = 25
\]
Thus, the new y-intercept is:
\[
(0, 25)
\]

### Summary of Intercepts:
- **Original Function**:
- x-intercepts: \( (5, 0) \) and \( (-5, 0) \)
- y-intercept: \( (0, -25) \)

- **Reflected Function**:
- x-intercepts: \( (5, 0) \) and \( (-5, 0) \) (no change)
- y-intercept: \( (0, 25) \) (changed from \( (0, -25) \) to \( (0, 25) \))

In conclusion, after reflecting over the x-axis, the x-intercepts remain unchanged, while the y-intercept changes from \( (0, -25) \) to \( (0, 25) \).