Bot GPT 4-o mini To determine which of the given functions will have the same x-intercept(s) as its reflection about the origin, we need to consider the reflection about the origin mathematically.
The reflection of a point \((x, y)\) about the origin is given by \((-x, -y)\). Therefore, for a function \(y = f(x)\), its reflection will be described by the function \(y = -f(-x)\).
Next, we need to find the x-intercepts of the original functions and their respective reflections:
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Function: \(y = 2x - 5\)
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Find x-intercept by setting \(y = 0\): \[ 0 = 2x - 5 \implies 2x = 5 \implies x = \frac{5}{2} \]
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Reflection: \(y = -f(-x) = - (2(-x) - 5) = -(-2x - 5) = 2x + 5\)
- Find x-intercept: \[ 0 = 2x + 5 \implies 2x = -5 \implies x = -\frac{5}{2} \]
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Conclusion: x-intercepts do not match.
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Function: \(y = x^2\)
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Find x-intercept by setting \(y = 0\): \[ 0 = x^2 \implies x = 0 \]
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Reflection: \(y = -f(-x) = -((-x)^2) = -x^2\)
- Find x-intercept: \[ 0 = -x^2 \implies x = 0 \]
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Conclusion: x-intercepts match.
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Function: \(y = (x - 4)^2\)
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Find x-intercept by setting \(y = 0\): \[ 0 = (x - 4)^2 \implies x - 4 = 0 \implies x = 4 \]
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Reflection: \(y = -f(-x) = -(-x - 4)^2 = -(x + 4)^2\)
- Find x-intercept: \[ 0 = -(x + 4)^2 \implies x + 4 = 0 \implies x = -4 \]
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Conclusion: x-intercepts do not match.
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Function: \(y = -\frac{1}{2}x + 6\)
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Find x-intercept by setting \(y = 0\): \[ 0 = -\frac{1}{2}x + 6 \implies \frac{1}{2}x = 6 \implies x = 12 \]
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Reflection: \(y = -f(-x) = -\left( -\frac{1}{2}(-x) + 6 \right) = -\left( \frac{1}{2}x + 6 \right) = -\frac{1}{2}x - 6\)
- Find x-intercept: \[ 0 = -\frac{1}{2}x - 6 \implies -\frac{1}{2}x = 6 \implies x = -12 \]
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Conclusion: x-intercepts do not match.
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Based on this analysis, the only function that has the same x-intercept as its reflection about the origin is:
\(y = x^2\).
1 answer