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Sagot :
Sure, let's break down the transformation of the graph of [tex]\( y = x^2 \)[/tex] to obtain the graph of [tex]\( y = -0.2x^2 \)[/tex].
### Step-by-Step Solution:
1. Starting with the Basic Graph:
The equation [tex]\( y = x^2 \)[/tex] represents a basic parabola that opens upwards. The vertex of this parabola is at the origin (0, 0), and it is symmetrical about the y-axis.
2. Reflection Across the x-Axis:
The negative sign in front of the coefficient [tex]\( -0.2 \)[/tex] indicates a reflection across the x-axis. This means that every point on the graph of [tex]\( y = x^2 \)[/tex] will be flipped over the x-axis, taking a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
As a result, the parabola will open downwards instead of upwards.
3. Vertical Compression:
The coefficient [tex]\( 0.2 \)[/tex] (which is a factor less than 1) indicates a vertical compression. This means that the graph of the parabola will be "compressed" towards the x-axis by this factor.
Specifically, in the function [tex]\( y = -0.2x^2 \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -0.2(1)^2 = -0.2 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = -0.2(2)^2 = -0.8 \)[/tex]
Compared to the original parabola [tex]\( y = x^2 \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = (1)^2 = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = (2)^2 = 4 \)[/tex]
We see that the values of [tex]\( y \)[/tex] in [tex]\( y = -0.2x^2 \)[/tex] are scaled down by a factor of 0.2 and then reflected (multiplied by -1).
### Description of the Graph of [tex]\( y = -0.2x^2 \)[/tex]:
The final graph is that of a parabola that opens downwards. It is a:
- Reflection across the x-axis of the graph of [tex]\( y = x^2 \)[/tex], because of the negative sign.
- Vertical compression by a factor of 0.2 compared to the graph of [tex]\( y = x^2 \)[/tex], meaning it is not as "tall" as the standard parabola.
### Conclusion:
The graph of [tex]\( y = -0.2x^2 \)[/tex] is a vertical compression by a factor of 0.2 and a reflection across the x-axis of the graph of [tex]\( y = x^2 \)[/tex].
### Step-by-Step Solution:
1. Starting with the Basic Graph:
The equation [tex]\( y = x^2 \)[/tex] represents a basic parabola that opens upwards. The vertex of this parabola is at the origin (0, 0), and it is symmetrical about the y-axis.
2. Reflection Across the x-Axis:
The negative sign in front of the coefficient [tex]\( -0.2 \)[/tex] indicates a reflection across the x-axis. This means that every point on the graph of [tex]\( y = x^2 \)[/tex] will be flipped over the x-axis, taking a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
As a result, the parabola will open downwards instead of upwards.
3. Vertical Compression:
The coefficient [tex]\( 0.2 \)[/tex] (which is a factor less than 1) indicates a vertical compression. This means that the graph of the parabola will be "compressed" towards the x-axis by this factor.
Specifically, in the function [tex]\( y = -0.2x^2 \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -0.2(1)^2 = -0.2 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = -0.2(2)^2 = -0.8 \)[/tex]
Compared to the original parabola [tex]\( y = x^2 \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = (1)^2 = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = (2)^2 = 4 \)[/tex]
We see that the values of [tex]\( y \)[/tex] in [tex]\( y = -0.2x^2 \)[/tex] are scaled down by a factor of 0.2 and then reflected (multiplied by -1).
### Description of the Graph of [tex]\( y = -0.2x^2 \)[/tex]:
The final graph is that of a parabola that opens downwards. It is a:
- Reflection across the x-axis of the graph of [tex]\( y = x^2 \)[/tex], because of the negative sign.
- Vertical compression by a factor of 0.2 compared to the graph of [tex]\( y = x^2 \)[/tex], meaning it is not as "tall" as the standard parabola.
### Conclusion:
The graph of [tex]\( y = -0.2x^2 \)[/tex] is a vertical compression by a factor of 0.2 and a reflection across the x-axis of the graph of [tex]\( y = x^2 \)[/tex].
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