To analyze the effect on the y-intercept when \( f(x) \) is replaced by \( af(x) \) with \( a \) being negative, we first need to determine the original y-intercept of \( f(x) = \sqrt{x} - 3 \).
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Finding the original y-intercept: To find the y-intercept, we substitute \( x = 0 \) into the function: \[ f(0) = \sqrt{0} - 3 = -3. \] Therefore, the original y-intercept is at the point \((0, -3)\).
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Transforming the function: Now consider the transformed function \( af(x) \): \[ af(x) = a(\sqrt{x} - 3). \] The new y-intercept can be found by substituting \( x = 0 \) again: \[ af(0) = a(\sqrt{0} - 3) = a(-3) = -3a. \]
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Effect of a being negative: Since \( a \) is negative, \( -3a \) will be a positive value when \( a \) is negative. This means the y-intercept moves from \(-3\) to a positive value that is further up the y-axis.
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Conclusion: The y-intercept shifts up by a distance of |a| \cdot 3 (which depends on the magnitude of \( a \)).
Among the given responses, the most accurate description (considering any generic negative \( a \)) is:
- The y-intercept shifts up by a distance of \( |3a| \) (where \( |a| \) is positive and the idea of shifting down can be interpreted based on the negative effect of \( a \)).
However, in terms of the options provided, the precise phrases might not all completely reflect the shifting caused by \( a \) without specifics on numerical values. Since \( a < 0 \) leads to some ambiguity in the provided options, a more generalized conclusion would be:
- The y-intercept shifts up (taking into account the magnitude of the negative value of \( a \)).
In summary, the key takeaway is that the y-intercept indeed shifts depending on the behavior of \( a \), specifically going to a positive intercept from a negative one as the original function shifts based on negative scaling.