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
Well..
They are same in such... they both open downwards
They are different in... their y-intercepts are different, their x-intercepts are different.
Once you have them graphed you will see the exact values :)
They are also the same in that they both have a maximum value at a their respective vertices : )
To graph the quadratic functions y = -2x^2 and y = -2x^2 + 4, we can start by creating a table of values to plot the points. We'll choose a range of x-values and substitute them into the equations to find the corresponding y-values.
Let's choose x-values from -3 to 3.
For y = -2x^2:
x = -3, y = -2(-3)^2 = -18
x = -2, y = -2(-2)^2 = -8
x = -1, y = -2(-1)^2 = -2
x = 0, y = -2(0)^2 = 0
x = 1, y = -2(1)^2 = -2
x = 2, y = -2(2)^2 = -8
x = 3, y = -2(3)^2 = -18
For y = -2x^2 + 4:
x = -3, y = -2(-3)^2 + 4 = 22
x = -2, y = -2(-2)^2 + 4 = 12
x = -1, y = -2(-1)^2 + 4 = 6
x = 0, y = -2(0)^2 + 4 = 4
x = 1, y = -2(1)^2 + 4 = 2
x = 2, y = -2(2)^2 + 4 = 0
x = 3, y = -2(3)^2 + 4 = -2
Now, we can plot these points on a graph.
For y = -2x^2, the graph is a downward-opening parabola that passes through the following points:
(-3, -18), (-2, -8), (-1, -2), (0, 0), (1, -2), (2, -8), (3, -18).
For y = -2x^2 + 4, the graph is also a downward-opening parabola, but it is shifted upward by 4 units compared to the first graph. It passes through the following points:
(-3, 22), (-2, 12), (-1, 6), (0, 4), (1, 2), (2, 0), (3, -2).
Comparing the two graphs, we see that they have the same shape, which is a downward-opening parabola. However, the graph of y = -2x^2 + 4 is shifted upwards by 4 units compared to the graph of y = -2x^2. This means the second graph is higher than the first graph at each corresponding x-value.
To graph the quadratic functions y = -2x^2 and y = -2x^2 + 4, you can use the following steps:
1. Choose a set of x-values to evaluate the functions. It is usually helpful to use a range of values that display the behavior of the functions. For example, you can choose x-values from -3 to 3.
2. Substitute each chosen x-value into the functions to find the corresponding y-values. For instance, for y = -2x^2, if x = -3, then y = -2(-3)^2 = -18. Repeat this process for all chosen x-values.
3. Plot the obtained points on a coordinate system. Use the x-values as the horizontal axis and the y-values as the vertical axis.
4. Connect the plotted points with a smooth curve. Since the functions are quadratic, the resulting graph will be a parabola.
Now let's compare and contrast the shape and position of the two graphs:
1. Shape: Both functions have the same general shape, which is a downward-opening parabola. This shape is characteristic of quadratic functions with a negative leading coefficient.
2. Position: The position of the graphs differ due to the constants in the function equations.
- For y = -2x^2, the graph is positioned at the origin (0,0). This is because there are no constants added or subtracted to the function.
- For y = -2x^2 + 4, the graph is shifted upward by 4 units compared to the first function. This is because the constant term 4 is added to the function.
Therefore, the graph of y = -2x^2 shows a downward-opening parabola centered at the origin, while the graph of y = -2x^2 + 4 is also a downward-opening parabola but is shifted upward by 4 units.
It is always helpful to visualize functions using a graph as it provides a clear understanding of their behavior and relationship.