Question
describe how the equation y=-3x^{2}, changes to parent graph
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GPT-4o mini
The equation \( y = -3x^2 \) is a transformation of the parent graph of the quadratic function \( y = x^2 \). Here's how it changes compared to the parent graph:
1. **Reflection**: The negative sign in front of the \( 3 \) indicates that the graph is reflected across the x-axis. This means that the parabola opens downwards instead of upwards.
2. **Vertical Stretch**: The coefficient \( -3 \) means that the parabola is vertically stretched by a factor of 3. This makes the graph narrower than the parent graph \( y = x^2 \). Instead of being "wider," the arms of the parabola are steeper, meaning that for any given value of \( x \), the value of \( y \) will be three times as far from the x-axis as it would be in the parent graph, but in the negative direction due to the reflection.
3. **Vertex**: The vertex of the parabola remains at the same point as the parent graph, which is at the origin (0, 0). There is no horizontal or vertical translation in this transformation.
To summarize, the transformation from the parent graph \( y = x^2 \) to \( y = -3x^2 \) involves reflecting the graph across the x-axis and vertically stretching it by a factor of 3, producing a narrower parabola that opens downwards.
1. **Reflection**: The negative sign in front of the \( 3 \) indicates that the graph is reflected across the x-axis. This means that the parabola opens downwards instead of upwards.
2. **Vertical Stretch**: The coefficient \( -3 \) means that the parabola is vertically stretched by a factor of 3. This makes the graph narrower than the parent graph \( y = x^2 \). Instead of being "wider," the arms of the parabola are steeper, meaning that for any given value of \( x \), the value of \( y \) will be three times as far from the x-axis as it would be in the parent graph, but in the negative direction due to the reflection.
3. **Vertex**: The vertex of the parabola remains at the same point as the parent graph, which is at the origin (0, 0). There is no horizontal or vertical translation in this transformation.
To summarize, the transformation from the parent graph \( y = x^2 \) to \( y = -3x^2 \) involves reflecting the graph across the x-axis and vertically stretching it by a factor of 3, producing a narrower parabola that opens downwards.
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