To determine which of the given functions will have the same x-intercept(s) as its reflection about the origin, we need to first understand what the reflection of a function about the origin means.
When a function \( y = f(x) \) is reflected about the origin, it transforms into \( y = -f(-x) \). The x-intercepts of a function are the points where \( y = 0 \). Therefore, we need to find the x-intercepts of both the original function and its reflection, and see if they are the same.
Let's analyze each function one by one.
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Function: \( y = 2x - 5 \)
- To find the x-intercept, set \( y = 0 \): \[ 0 = 2x - 5 \implies 2x = 5 \implies x = \frac{5}{2} \]
- Reflecting the function yields \( y = -(2(-x) - 5) = -(-2x - 5) = 2x + 5 \).
- To find the x-intercept of the reflection: \[ 0 = 2x + 5 \implies 2x = -5 \implies x = -\frac{5}{2} \]
- The x-intercepts are different.
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Function: \( y = x^2 \)
- Finding the x-intercept: \[ 0 = x^2 \implies x = 0 \]
- Reflecting the function yields \( y = -(-x)^2 = -x^2 \).
- The x-intercept of the reflection: \[ 0 = -x^2 \implies x^2 = 0 \implies x = 0 \]
- The x-intercepts are the same.
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Function: \( y = (x - 4)^2 \)
- Finding the x-intercept: \[ 0 = (x - 4)^2 \implies x - 4 = 0 \implies x = 4 \]
- Reflecting the function yields \( y = -((-x) - 4)^2 = -(-x - 4)^2 \).
- The x-intercept of the reflection: \[ 0 = -(x + 4)^2 \implies (x + 4)^2 = 0 \implies x = -4 \]
- The x-intercepts are different.
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Function: \( y = -\frac{1}{2}x + 6 \)
- Finding the x-intercept: \[ 0 = -\frac{1}{2}x + 6 \implies \frac{1}{2}x = 6 \implies x = 12 \]
- Reflecting the function yields \( y = -\left(-\frac{1}{2}(-x) + 6\right) = -\left(\frac{1}{2}x + 6\right) = -\frac{1}{2}x - 6 \).
- The x-intercept of the reflection: \[ 0 = -\frac{1}{2}x - 6 \implies -\frac{1}{2}x = 6 \implies x = -12 \]
- The x-intercepts are different.
In summary, the only function that maintains the same x-intercept as its reflection about the origin is:
\( y = x^2 \).