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Question Graph the quadratic functions y = -2x2 and y = -2x2 + 4 on a separate piece of paper. Using those graphs, compare and contrast the shape and position of the graphs.
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GPT 3.5
To graph the quadratic functions y = -2x^2 and y = -2x^2 + 4, we can first identify the vertex of the parabolas using the formula x = -b/2a.
For y = -2x^2, a = -2 and b = 0, so the vertex is at x = 0. Substituting x = 0 into the equation, we find the y-coordinate of the vertex to be y = 0.
For y = -2x^2 + 4, a = -2 and b = 0, so the vertex is also at x = 0. Substituting x = 0 into the equation, we find the y-coordinate of the vertex to be y = 4.
Now we can plot the vertex of each parabola and determine the shape of the curve. Since the coefficient of x^2 is negative in both equations, the parabolas will open downwards.
The graph of y = -2x^2 will be a downward-facing parabola with vertex at (0, 0), while the graph of y = -2x^2 + 4 will be a downward-facing parabola with vertex at (0, 4).
Comparing the two graphs:
- Both parabolas have the same shape, opening downwards.
- The graph of y = -2x^2 is below the x-axis and does not intersect the y-axis, while the graph of y = -2x^2 + 4 is shifted upwards by 4 units and intersects the y-axis at y = 4.
Overall, the two graphs are similar in shape but have different positions due to the vertical shift in the second equation.
For y = -2x^2, a = -2 and b = 0, so the vertex is at x = 0. Substituting x = 0 into the equation, we find the y-coordinate of the vertex to be y = 0.
For y = -2x^2 + 4, a = -2 and b = 0, so the vertex is also at x = 0. Substituting x = 0 into the equation, we find the y-coordinate of the vertex to be y = 4.
Now we can plot the vertex of each parabola and determine the shape of the curve. Since the coefficient of x^2 is negative in both equations, the parabolas will open downwards.
The graph of y = -2x^2 will be a downward-facing parabola with vertex at (0, 0), while the graph of y = -2x^2 + 4 will be a downward-facing parabola with vertex at (0, 4).
Comparing the two graphs:
- Both parabolas have the same shape, opening downwards.
- The graph of y = -2x^2 is below the x-axis and does not intersect the y-axis, while the graph of y = -2x^2 + 4 is shifted upwards by 4 units and intersects the y-axis at y = 4.
Overall, the two graphs are similar in shape but have different positions due to the vertical shift in the second equation.
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