The parent function \( f(x) = x^2 \) has a basic upward "U" shape. When we look at the points provided: \( (0,0) \), \( (1,9) \), and \( (-1,9) \), we can analyze how the function has been transformed.
The original points of the parent function are \( (0,0) \), \( (1,1) \), and \( (-1,1) \). In our case, the points at \( x = 1 \) and \( x = -1 \) yield \( f(1) = 9 \) and \( f(-1) = 9 \), which suggests that the output has increased significantly.
To achieve this, the function must have undergone a vertical dilation (stretch). Specifically, it appears that the function has been transformed by multiplicative factor. Let's rewrite the function based on the new points. The relationship shows that both points now yield \( 9 \) instead of \( 1 \), specifically:
\[ f(1) = k(1^2) = 9, \text{ which gives us } k = 9. \]
Therefore, the transformed function can be expressed as:
\[ g(x) = 9x^2. \]
So, the dilation that has occurred is a vertical stretch by a factor of \( 9 \) (stretching upwards), which explains why the points \( (1,9) \) and \( (-1,9) \) are much higher than they would normally be in the parent function \( f(x) = x^2 \).
Among the options provided:
- Vertically translated down: No, it has not been translated down.
- Horizontally stretched: No, there is no indication it has been stretched horizontally.
- Horizontally compressed: No, there is no evidence of horizontal compression.
- Vertically translated up: While the graph does appear higher, it is specifically due to dilation rather than translation.
Thus, the correct explanation is that the graph has undergone a vertical stretch, although it is not explicitly listed as an option. The key takeaway is that the output values have been scaled up vertically by a factor, resulting in the points observed.
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