which steps transform the graph of y=x^2 to y=-2(x-2)^2+2
The steps to transform the graph of y = x^2 to y = -2(x-2)^2 + 2 are:
1. Reflect the graph vertically: Multiply the original equation by -1 to reflect the graph vertically. This changes y = x^2 to y = -x^2.
2. Stretch the graph vertically: Multiply the equation by a constant to stretch or compress the graph vertically. In this case, multiply y = -x^2 by -2 to stretch the graph vertically. The equation becomes y = -2x^2.
3. Move the graph horizontally: Add or subtract a value inside the parentheses to shift the graph horizontally. In this case, subtract 2 inside the parentheses to move the graph 2 units to the right. The equation becomes y = -2(x-2)^2.
4. Move the graph vertically: Add or subtract a value outside the parentheses to shift the graph vertically. In this case, add 2 outside the parentheses to move the graph 2 units up. The final equation becomes y = -2(x-2)^2 + 2.
To transform the graph of y = x^2 to y = -2(x-2)^2 + 2, you need to perform the following steps:
Step 1: Translation (Horizontal)
- Since the expression (x-2) appears inside the parentheses, it indicates a horizontal translation.
- To move the graph 2 units to the right, we shift all the x-values (points) of the original graph to the right by 2 units.
Step 2: Reflection (Vertical)
- The negative sign in front of the 2 in the transformed equation indicates a vertical reflection.
- By reflecting the graph, all the y-values of the original graph will be negated.
Step 3: Stretch/Compression (Vertical)
- The number 2 inside the parenthesis indicates a vertical stretch or compression compared to the original graph.
- Since the coefficient is 2, it indicates a vertical compression.
- This means that the graph will be narrower than the original graph.
Step 4: Vertical Translation
- The constant term +2 at the end of the equation indicates a vertical shift up by 2 units.
- All the y-values of the transformed graph will be increased by 2 units.
By following these steps, you can transform the graph of y = x^2 to y = -2(x-2)^2 + 2.
To transform the graph of y = x^2 to y = -2(x - 2)^2 + 2, you need to follow these steps:
Step 1: Vertical translation
The term -2(x - 2)^2 moves the entire graph 2 units upward. The "+2" at the end will shift the graph vertically upwards by 2 units.
Step 2: Horizontal translation
The term (x - 2) will shift the graph horizontally to the right by 2 units. Since it is a positive value, the shift will be in the positive x-direction.
Step 3: Vertical transformation
The term -2 will stretch the graph vertically by a factor of 2. Since this value is negative, the graph will be reflected about the x-axis.
By applying these transformations step by step, you can transform the original graph of y = x^2 to its new form, y = -2(x - 2)^2 + 2.