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?

Rationale

y = - 3x ^ 2 - 5

The final equation after applying the transformations to the graph of y = x^2 is y = -3x^2 - 5. This results from stretching the graph vertically by a factor of 3, translating it upward by 5 units, and reflecting it across the x-axis.

A (y = - 3x ^ 2 - 5) CORRECT

This equation correctly reflects all the transformations applied to the original graph. The vertical stretch by a factor of 3 is represented by -3 in front of x^2, the upward translation of 5 units is represented by -5, and the reflection across the x-axis is indicated by the negative sign in front of the 3.

B (y = - 3x ^ 2 + 5) YOUR ANSWER

While this equation includes the vertical stretch and reflection, it incorrectly indicates an upward translation of 5 units, rather than downward. The +5 would suggest the graph is 5 units above the x-axis, which contradicts the translation specified in the problem.

C (y = - 3 * (x - 5) ^ 2) YOUR ANSWER

This equation suggests a horizontal translation of 5 units to the right, rather than the correct upward translation. The negative coefficient reflects the graph across the x-axis, but the incorrect horizontal shift alters the graph's position in the xy-plane.

D (y = - 3 * (x + 5) ^ 2) YOUR ANSWER

Similar to option C, this equation also indicates a horizontal translation, but this time to the left by 5 units. The vertical stretch and reflection are correctly represented, but the horizontal shift is incorrect, preventing it from matching the final graph.

E (y = 3x ^ 2 – 5) YOUR ANSWER

This equation fails to reflect across the x-axis, as it lacks a negative sign before the vertical stretch factor. It also indicates a downward shift of 5 units, which is contrary to the upward translation that was specified.

Conclusion

The sequence of transformations applied to the graph of y = x^2 results in the final equation y = -3x^2 - 5. This accurately incorporates the vertical stretch, upward translation, and reflection across the x-axis. All other options either misrepresent one of these transformations or incorrectly shift the graph horizontally, leading to incorrect equations.

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?

y = - 3x ^ 2 - 5

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