Describe the following transformations on y=x^2:
y=-4(2x+10)^2 -7
the -4 and the x-coefficient cause the parabola to open downward and "stretch out" in the y direction
the 10 and the -7 move the vertex away from the origin
Make a preliminary sketch of y = x^2
y=-4(2x+10)^2 -7
= -4(2^2)(x+5)^2 - 7
= -16(x+5)^2 - 7
The vertex of y = x^2, which was (0,0) is now (-5,-7), the direction is now downwards, the parabola is stretched by a factor of 16, which makes it much more "pointed"
To describe the transformations applied to the function y = x^2 in the equation y = -4(2x + 10)^2 - 7, we break it down step-by-step:
1. Horizontal Translation: The expression inside the parentheses, 2x + 10, indicates a horizontal translation of -10 units to the left. This means the graph is shifted to the left by 10 units.
2. Horizontal Scaling: The coefficient 2 in front of the x causes a horizontal scaling. Normally, the graph of y = x^2 is wider, but with the coefficient of 2, it becomes narrower. Specifically, 2x + 10 is compressed horizontally by a factor of 2.
3. Vertical Scaling: The coefficient -4 outside the parentheses causes a vertical scaling by a factor of -4. This means the graph is flipped upside down and vertically stretched by a factor of 4.
4. Vertical Translation: Finally, the constant -7 at the end causes a vertical translation of 7 units downwards. This means the graph is shifted 7 units down.
In summary, the transformations applied to y = x^2 in y = -4(2x + 10)^2 - 7 are:
- Horizontal translation by 10 units to the left.
- Horizontal scaling by a factor of 2, making the graph narrower.
- Vertical scaling by a factor of -4, flipping it upside down and stretching it vertically.
- Vertical translation by 7 units downwards.
To describe the transformations on the equation y = x^2 into y = -4(2x+10)^2 -7, we can break down the transformations step by step.
1. Horizontal translation: The expression (2x + 10) inside the parentheses represents a horizontal translation. Specifically, 2x shifts the graph 2 units to the right, and +10 shifts it an additional 10 units to the left. So, there is a horizontal translation of 10 units to the left.
2. Vertical reflection: The negative sign in front of the entire equation, -4(2x+10)^2 -7, indicates a vertical reflection. This means the graph is being flipped over the x-axis. Instead of opening upward, the parabola opens downward.
3. Vertical compression: The coefficient -4 in front of the equation represents vertical compression. It indicates that the graph is compressed vertically by a factor of 4 compared to the original y = x^2 graph. This means the graph becomes narrower and steeper.
4. Vertical shift: The constant term -7 at the end of the equation represents a vertical shift downward. The graph is shifted 7 units downward compared to the original y = x^2 graph.
To summarize, the steps to transform the graph of y = x^2 into y = -4(2x+10)^2 -7 are:
1. Horizontal translation 10 units to the left.
2. Vertical reflection (flipping over the x-axis).
3. Vertical compression by a factor of 4.
4. Vertical shift downward 7 units.