Which steps transform the graph of y = x^2 to y = -2(x-2)^2 + 2? (1 point)
A. translate 2 units to the left, translate down 2 units, stretch by the factor 2
B. translate 2 units to the right, translate up 2 units, stretch by the factor 2
C. reflect across the x-axis, translate 2 units to the left, translate down 2 units, stretch by the factor 2
D. reflect across the x-axis, translate 2 units to the right, translate up 2 units, stretch by the factor 2
The correct answer is D.
To transform the graph of y = x^2 to y = -2(x-2)^2 + 2, the following steps need to be taken:
1. Reflect across the x-axis: This will change the sign of y, so the graph will now be y = -x^2.
2. Translate 2 units to the right: This means that the graph will be shifted horizontally to the right by 2 units, resulting in y = -(x-2)^2.
3. Translate up 2 units: This means that the graph will be shifted vertically upwards by 2 units, resulting in y = -(x-2)^2 + 2.
4. Stretch by the factor 2: This means that the graph will be vertically stretched by a factor of 2, resulting in y = -2(x-2)^2 + 2.
Therefore, the correct steps to transform the graph of y = x^2 to y = -2(x-2)^2 + 2 are to reflect across the x-axis, translate 2 units to the right, translate up 2 units, and finally stretch by the factor 2, which is answer D.
To transform the graph of y = x^2 to y = -2(x-2)^2 + 2, we need to apply a series of steps. Let's go through each option and determine which steps are involved.
Option A: Translate 2 units to the left, translate down 2 units, stretch by a factor of 2.
To translate 2 units to the left, we replace the variable x with (x + 2). This results in the equation y = (x + 2)^2. However, this option specifies a translation to the left, so it doesn't match.
Option B: Translate 2 units to the right, translate up 2 units, stretch by a factor of 2.
To translate 2 units to the right, we replace the variable x with (x - 2), resulting in y = (x - 2)^2. This matches the first part of the equation y = -2(x-2)^2 + 2. However, no stretching is involved in this transformation, so it doesn't fully match.
Option C: Reflect across the x-axis, translate 2 units to the left, translate down 2 units, stretch by a factor of 2.
A reflection across the x-axis is achieved by multiplying the entire equation by -1, resulting in y = -x^2. To translate 2 units to the left, we replace the variable x with (x + 2), yielding y = -(x + 2)^2. Additionally, a translation down 2 units can be achieved by subtracting 2 from the equation, which gives y = -((x + 2)^2 - 2).
So far, the equation matches the form y = -2(x-2)^2 but is missing the "+ 2" at the end. Lastly, there is no mention of a stretch by a factor of 2, so this option doesn't fully match either.
Option D: Reflect across the x-axis, translate 2 units to the right, translate up 2 units, stretch by a factor of 2.
A reflection across the x-axis gives us y = -x^2. To translate 2 units to the right, we replace the variable x with (x - 2), resulting in y = -(x - 2)^2. Next, a translation up 2 units can be achieved by adding 2 to the equation, giving us y = -(x - 2)^2 + 2.
So far, the equation matches the form y = -2(x-2)^2 but is missing the "-2" in front. Lastly, this option specifies a stretch by the factor of 2, which matches the given equation.
Therefore, the correct option is D: Reflect across the x-axis, translate 2 units to the right, translate up 2 units, stretch by a factor of 2.