153k views
5 votes
What steps transform the graph of y = x^2 to y = -2(x - 3)^2 + 5?

a) Horizontal shift right by 3 units, vertical stretch by a factor of 2, vertical shift up by 5 units.
b) Horizontal shift left by 3 units, vertical stretch by a factor of 2, vertical shift up by 5 units.
c) Horizontal shift right by 3 units, vertical compression by a factor of 2, vertical shift up by 5 units.
d) Horizontal shift left by 3 units, vertical compression by a factor of 2, vertical shift up by 5 units.

User Emad Amien
by
7.4k points

1 Answer

3 votes

Final answer:

To transform the graph of y = x^2 to y = -2(x - 3)^2 + 5, we apply a horizontal shift to the right by 3 units, a reflection over the x-axis combined with a vertical stretch by a factor of 2, and a vertical shift upwards by 5 units. This corresponds to option (a).

Step-by-step explanation:

To transform the graph of y = x^2 to y = -2(x - 3)^2 + 5, let's analyze the equation step by step:

  • The (x - 3) inside the parentheses indicates a horizontal shift to the right side of the coordinate system by 3 units, because we are effectively saying where the graph used to be 'zeroed' at x=0, it is now 'zeroed' at x=3.
  • The leading negative sign in front of the 2 indicates a reflection over the x-axis, which is a vertical flip that causes our parabola which opened upwards to now open vertically downward in the coordinate system.
  • The coefficient of 2, which is the constant multiplier of the squared term, increases the steepness of the parabola, which is not just a stretch but also involves the reflection due to the negative sign. In absence of the negative, this would be a vertical stretch by a factor of 2; with the negative, it's a vertical stretch combined with a vertical flip.
  • Finally, the +5 outside the parentheses indicates a vertical shift upwards by 5 units in the coordinate system.

So, the correct sequence to transform y = x^2 into y = -2(x - 3)^2 + 5 is a horizontal shift to the right by 3 units, reflection and vertical stretch by a factor of 2 (due to multiplication by -2), and a vertical shift upwards by 5 units.

This corresponds to option (a), as it fully describes the transformations applied.

User KeuleJ
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.