Graph the quadratic functions y = -2x^2 and y = -2x^2 + 4 on a separate piece of paper. Using those graphs, compare and contrast the shape and position of the graphs.
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If you look at the functions, you can see that the 2nd is exactly the same as the first, but 4 units higher (that +4 at the end)
Now, assuming you can graph a parabola, the rest is easy.
To graph the quadratic functions on a separate piece of paper, we can follow these steps:
1. Create a set of coordinate axes. Draw a horizontal x-axis and a vertical y-axis, intersecting at the origin (0,0).
2. For the first function, y = -2x^2, we can start by finding some key points to plot. To do this, select a few x-values and calculate their corresponding y-values using the equation. For example, you could choose x = -2, -1, 0, 1, and 2.
When x = -2: y = -2(-2)^2 = -2(4) = -8.
When x = -1: y = -2(-1)^2 = -2(1) = -2.
When x = 0: y = -2(0)^2 = 0.
When x = 1: y = -2(1)^2 = -2(1) = -2.
When x = 2: y = -2(2)^2 = -2(4) = -8.
Plot these points on your graph, yielding the points (-2, -8), (-1, -2), (0, 0), (1, -2), and (2, -8).
3. Connect the plotted points with a smooth curve. Since the equation indicates a quadratic function with a negative leading coefficient (-2), the graph will open downward.
Now, let's move on to the second function, y = -2x^2 + 4.
4. Follow the same process as before to find key points. Using the same x-values (x = -2, -1, 0, 1, and 2):
When x = -2: y = -2(-2)^2 + 4 = -2(4) + 4 = -8 + 4 = -4.
When x = -1: y = -2(-1)^2 + 4 = -2(1) + 4 = -2 + 4 = 2.
When x = 0: y = -2(0)^2 + 4 = 0 + 4 = 4.
When x = 1: y = -2(1)^2 + 4 = -2(1) + 4 = -2 + 4 = 2.
When x = 2: y = -2(2)^2 + 4 = -2(4) + 4 = -8 + 4 = -4.
Plot these points on the same graph, resulting in the points (-2, -4), (-1, 2), (0, 4), (1, 2), and (2, -4).
5. Connect these points with a smooth curve. Since the quadratic equation has the same form as the previous one, the shape will be identical. However, the graph is vertically shifted upward by 4 units due to the +4 constant term.
Now, let's compare and contrast the shape and position of the two graphs:
- Shape: Both graphs have a downward-opening parabolic shape since the leading coefficient is negative. They are symmetrical about the vertical axis.
- Position: The graph of y = -2x^2 is positioned completely below the x-axis since there is no vertical shift. On the other hand, the graph of y = -2x^2 + 4 is shifted upwards by four units due to the constant term +4. It intersects the y-axis at the point (0, 4).
By sketching both graphs, you'll be able to visually compare and contrast the shape and position more effectively.