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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To graph the quadratic functions y = -2x^2 and y = -2x^2 + 4, you can follow these steps:

1. Determine the shape of the graph:
The general form of a quadratic function is y = ax^2 + bx + c. If the coefficient "a" is positive, the graph opens upward like a U-shape, known as a "concave up" graph. If "a" is negative, the graph opens downward like an upside-down U-shape, known as a "concave down" graph. In both cases, the vertex of the parabola is a significant point.

2. Find the vertex of the parabolas:
For a quadratic function in the form y = ax^2 + bx + c, the x-coordinate of the vertex is given by x = -b / (2a). By substituting this x-value into the equation, you can find the y-coordinate of the vertex.

3. Plot the vertex on the graph:
Draw a point on the x-axis at the computed x-coordinate and its corresponding y-coordinate on the y-axis. This represents the vertex of the parabola.

4. Determine additional points:
Select a few values for x (both positive and negative) and plug them into the equation to find the corresponding y-values. This will give you other points on the graph.

5. Plot the points and draw the graph:
Connect the plotted points using a smooth curve to create the graph of the quadratic function.

Now let's apply these steps to the given quadratic functions:

For y = -2x^2:

1. Shape: Since the coefficient "a" is negative (-2), the graph will open downward, forming a concave down parabola.

2. Vertex: To find the vertex, use the formula x = -b / (2a). Since b = 0 and a = -2, the x-coordinate of the vertex is x = 0 / (-4) = 0. Substitute x = 0 into the equation to find the y-coordinate: y = -2(0)^2 = 0. Therefore, the vertex is (0, 0).

3. Additional points: Choose a few values for x such as x = -2, -1, 1, and 2, and calculate the corresponding y-values: (-2, -8), (-1, -2), (1, -2), (2, -8).

4. Plotting the points and drawing the graph:

-2 | * y = -2x^2
| *
| *
| *
| *
|.____________________.
| -2 | -1 | 0 | 1 | 2 | x

For y = -2x^2 + 4:

1. Shape: With the same coefficient "a" (-2), the shape of the graph remains the same as y = -2x^2, opening downward in a concave down manner.

2. Vertex: As before, use the formula x = -b / (2a). Since b = 0 and a = -2, the x-coordinate of the vertex is x = 0 / (-4) = 0. Find the y-coordinate: y = -2(0)^2 + 4 = 4. Thus, the vertex is (0, 4).

3. Additional points: Following the same steps, substitute values for x (e.g., -2, -1, 1, and 2) to obtain the corresponding y-values: (-2, 0), (-1, 2), (1, 2), (2, 0).

4. Plotting the points and drawing the graph:

-2 | * y = -2x^2 + 4
| *
| *
| * +
| *
|.____________________.
| -2 | -1 | 0 | 1 | 2 | x

By comparing the two graphs, you can see that they have the same concave down shape (the left side of a U-shape), but they are vertically shifted relative to each other. The graph of y = -2x^2 is located entirely below the x-axis, while the graph of y = -2x^2 + 4 is shifted upward by 4 units, intersecting the y-axis at y = 4.