10. In the xy-plane, the graph of the equation y = x ^ 2 is stretched by a factor of 3 in the vertical direction. The resulting graph is then translated 5 units upward, and finally the translated graph is reflected across the x-axis. Which of the following is an equation of the final result?
Answer: A
y = - 3x ^ 2 - 5
The final equation of the graph after performing the transformations is y = -3x^2 - 5. This equation reflects the vertical stretch, upward translation, and reflection across the x-axis.
A) y = - 3x ^ 2 - 5
This option accurately represents the final transformation sequence. The original equation y = x^2 is stretched vertically by a factor of 3 to yield y = 3x^2, then translated upward by 5 units to become y = 3x^2 + 5. Finally, reflecting across the x-axis results in the equation y = -3x^2 - 5.
B) y = - 3x ^ 2 + 5
This option is incorrect because, while it reflects the vertical stretch by a factor of 3, it does not account for the reflection across the x-axis. The upward translation of 5 units should be negative after reflection, which is not represented here.
C) y = - 3 * (x - 5) ^ 2
This option is incorrect as it alters the x-term in a way that indicates a horizontal shift rather than a vertical transformation. The reflection and translation processes have not been applied correctly, leading to a fundamentally different graph.
D) y = - 3 * (x + 5) ^ 2
This option is also incorrect. Similar to Option C, it modifies the x-term incorrectly, implying a horizontal shift. This does not reflect the transformations described in the problem statement, as it does not follow the correct vertical stretch and reflection.
E) y = 3x ^ 2 – 5
This option is incorrect because it fails to reflect the graph across the x-axis. While it correctly shows the vertical stretch, the upward translation should be negative after the reflection, which is not represented here.
Conclusion
The correct answer, y = -3x^2 - 5, reflects the required transformations accurately: the vertical stretch by a factor of 3, upward translation of 5 units, and reflection across the x-axis. All other options either misrepresent the transformations or incorrectly modify the x-term, failing to achieve the desired final result.