Which steps transform the graph of y=x^(2) to y=-2(x-2)^(2)+2
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. translate 2 units to the right, reflect across the x-axis, stretch by the factor 2, and translate up 2 units
The correct answer is D. translate 2 units to the right, reflect across the x-axis, stretch by the factor 2, and translate up 2 units.
Let's go through each step:
1. Translate 2 units to the right: This means we shift the graph 2 units to the right. This is indicated by the (x-2) in the equation.
2. Reflect across the x-axis: The negative sign in front of the equation indicates a reflection across the x-axis. This means the graph is flipped upside down.
3. Stretch by a factor of 2: The coefficient -2 in front of the equation indicates a vertical stretch by a factor of 2. This means the graph is stretched vertically, making it narrower.
4. Translate up 2 units: This means we shift the graph 2 units upward. This is indicated by the "+2" at the end of the equation.
All these transformations result in the graph of y=-2(x-2)^(2)+2 which is a transformation of the original graph y=x^(2).
To transform the graph of y = x^2 to y = -2(x-2)^2 + 2, we need to follow a series of steps. Let's analyze each option to determine the correct steps:
A. Translate 2 units to the left, translate down 2 units, stretch by the factor 2:
This option involves shifting the graph to the left and downward. However, it does not specify anything about the y-axis. Since we have a negative coefficient in front of the equation, we require a reflection across the x-axis. Therefore, this option is incorrect.
B. Translate 2 units to the right, translate up 2 units, stretch by the factor 2:
This option involves shifting the graph to the right and upward. However, it does not mention anything about reflecting the graph across the x-axis. As we require a reflection, this option is also incorrect.
C. Reflect across the x-axis, translate 2 units to the left, translate down 2 units, stretch by the factor 2:
This option includes all the necessary steps. Reflecting across the x-axis accounts for the negative coefficient in the equation. Then, we translate the graph two units to the left and two units down. Finally, we stretch the graph by a factor of 2. Therefore, this option is the correct answer.
D. Translate 2 units to the right, reflect across the x-axis, stretch by the factor 2, and translate up 2 units:
This option involves the correct reflection across the x-axis and stretching by a factor of 2. However, it does not include the necessary translation of 2 units to the left. Hence, this option is incorrect.
In conclusion, the correct answer is option C: reflect across the x-axis, translate 2 units to the left, translate down 2 units, stretch by the factor 2.
The correct answer is D.
D. translate 2 units to the right, reflect across the x-axis, stretch by the factor 2, and translate up 2 units.
Let's break down the steps:
1. Translate 2 units to the right: This means shifting the graph horizontally to the right by 2 units. In this case, the x-coordinate of every point on the graph is increased by 2.
2. Reflect across the x-axis: This means flipping the graph upside down. The positive y-values become negative, and the negative y-values become positive.
3. Stretch by the factor 2: This means multiplying the y-values by 2 to vertically stretch the graph. The x-values remain the same.
4. Translate up 2 units: Finally, this step shifts the graph vertically upward by 2 units. The y-coordinate of every point on the graph is increased by 2.
By following these steps in order, you will transform the graph of y = x^2 to y = -2(x - 2)^2 + 2.