To interpret the transformation of the given function \( y = \frac{3}{\sqrt{x}} \), we'll begin with its original form and analyze the provided points.
-
Original function: The original function \( y = \frac{3}{\sqrt{x}} \) has a vertical asymptote at \( x = 0 \) and a shape that approaches the x-axis as \( x \) increases.
-
Key features of the function:
- When \( x = 1 \), \( y = 3 \).
- When \( x = 4 \), \( y = \frac{3}{2} \).
- As \( x \to 0^+ \), \( y \to \infty \).
- As \( x \) increases, \( y \) decreases.
-
Given points: The points provided are:
- (-1, -3)
- (0, -2)
- (1, 5)
- (2, ?)
Firstly, we notice some of these points have either negative x-values or x-values that are undefined for the original function \( y = \frac{3}{\sqrt{x}} \) because the function only exists for \( x > 0 \).
Based on this information, let's evaluate the possible transformations:
Evaluating the Transformations
-
Shift Left or Right:
- The function does not have defined values for negative \( x \). Therefore, there cannot be a simple left or right shift.
-
Shift Up or Down:
- If a value of \( y \) in a function increases or decreases from its original form, it indicates a vertical shift.
- For example, translating a function down would involve replacing \( y \) with \( y - k \) (where \( k \) is a positive number).
Points Interpretation
Since some of the provided points seem to be inconsistent with the original function's domain and behavior, it suggests a possible vertical transformation.
Based on what you specified:
- The graph might have shifted down because of some y-values' altitudes and the mention of the change in y-intercept.
- For instance, our original function at \( x = 1 \) produces \( 3 \), while the axis indicates a drop with points being lower than that.
Conclusion
From this analysis, the most reasonable transformation indicated by your points is:
The graph represents a shift down due to the change in the y-intercept.
If you meant that the function was altered to one that could accept values in the negative domain, further clarifications on the function's new form would be necessary to denote the specific transformation. However, the interpretation suggests a vertical shift downwards based on expected behavior and points provided.