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.
As an AI language model, I cannot draw graphs, but I can explain.
The quadratic functions y = -2x2 and y = -2x2 + 4 have the same shape, which is a downward-facing parabola. The negative coefficient of x2 tells us that the parabolas open downwards.
The vertex of both parabolas is at (0,0), which means that they are symmetric about the y-axis.
However, the two graphs differ in their y-intercepts. The graph of y = -2x2 crosses the y-axis at 0, while the graph of y = -2x2 + 4 is shifted up by 4 units and crosses the y-axis at (0,4).
Therefore, the second parabola is simply a vertical shift of the first one.
To graph the quadratic functions y = -2x^2 and y = -2x^2 + 4, follow these steps:
1. Choose a set of x-values. Start by selecting a range of x-values that will give a good representation of the shape of the graph. For example, you can choose values like -2, -1, 0, 1, and 2.
2. Calculate the corresponding y-values. Plug each x-value into the function to find the corresponding y-values. For y = -2x^2, multiply each x-value by itself and then multiply the result by -2. For y = -2x^2 + 4, calculate the y-values similarly and then add 4 to each result.
3. Plot the points on a graph. Use the x and y-values to plot the points on the graph paper. Do this for both functions.
4. Connect the points. Once all the points are plotted, draw a smooth curve that passes through all the points for each function. Remember that the graph of a quadratic function is a parabola.
Now, let's compare and contrast the shape and position of the graphs:
- The shape: Both graphs have a similar shape, which is a downward-opening parabola. This is because the coefficient of x^2 in both functions is negative (-2).
- The position: The position of the two graphs differs in terms of their y-intercepts. The graph of y = -2x^2 intersects the y-axis at the origin (0, 0), while the graph of y = -2x^2 + 4 intersects the y-axis at the point (0, 4). This means that the second graph has been shifted upward by 4 units compared to the first graph.
- The vertex: The vertex represents the lowest point of the parabola. For both functions, the vertex is located at the point (0, 0), as the coefficient of x in both cases is 0.
- The range: The range of y-values for both functions is (-∞, 0]. This means that the value of y is always less than or equal to 0 for all x-values.
By comparing and contrasting the shape and position of the graphs, you can clearly see that the two functions have the same shape, but they differ in terms of their vertical position (y-intercept) on the graph.