To analyze the function \( y = 4 - \sqrt{2x} \), let's go through each statement one by one.
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The x-intercept of the graph is (4, 0).
- To find the x-intercept, we set \( y = 0 \): \[ 0 = 4 - \sqrt{2x} \] Rearranging gives \( \sqrt{2x} = 4 \). Squaring both sides yields \( 2x = 16 \), or \( x = 8 \). Thus, the x-intercept is \( (8, 0) \), not (4, 0). So, this statement is false.
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The graph has no y-intercepts.
- To find the y-intercept, we set \( x = 0 \): \[ y = 4 - \sqrt{2 \cdot 0} = 4 - 0 = 4. \] The y-intercept is \( (0, 4) \). Therefore, this statement is false.
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The maximum of the graph occurs at the y-intercept.
- The graph \( y = 4 - \sqrt{2x} \) decreases as \( x \) increases. The maximum value occurs at the leftmost point, which is the y-intercept \( (0, 4) \). This statement is true.
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The minimum of the graph occurs at the x-intercept.
- The x-intercept is \( (8, 0) \), which is indeed a minimum point of the graph since the function continues to decrease after this point. However, we typically think of the minimum value in the context of the entire range of the function as \( y \) can approach \( -\infty \) as \( x \) approaches larger values. This statement can be misleading as minimum is commonly referring to the lowest output that the function can reach. Therefore, while this statement isn’t strictly incorrect, it doesn't capture the broader behavior of the function.
In conclusion, the accurate statement is:
The maximum of the graph occurs at the y-intercept.