Use the graph of y=x^2 to graph
G(x)=x^2+4
G(x)=(x-4)^2
To graph G(x) = x^2 + 4, we start with the graph of y = x^2.
Here is the graph of y = x^2:
The graph of y = x^2 is a parabola that opens upwards and has its vertex at the origin (0,0). It passes through the points (-2, 4), (-1, 1), (0, 0), (1, 1), and (2, 4).
Now, let's graph G(x) = x^2 + 4. To obtain G(x), we shift the graph of y = x^2 upwards by 4 units.
To shift the graph upwards, we add 4 to each y-coordinate of the original parabola:
The graph of G(x) = x^2 + 4 is also a parabola that opens upwards, but it is shifted 4 units above the graph of y = x^2. The vertex of G(x) is located at (0, 4) and it passes through the points (-2, 8), (-1, 5), (0, 4), (1, 5), and (2, 8).
Next, let's graph G(x) = (x-4)^2. To obtain G(x), we shift the graph of y = x^2 horizontally 4 units to the right.
To shift the graph to the right, we replace x with (x-4) in the equation of the original parabola:
The graph of G(x) = (x-4)^2 is a parabola that opens upwards and is shifted 4 units to the right compared to the graph of y = x^2. The vertex of G(x) is located at (4, 0) and it passes through the points (0, 16), (1, 9), (2, 4), (3, 1), and (4, 0).
Here is the final graph of G(x) = (x-4)^2:
The graph of G(x) = (x-4)^2 is a parabola that opens upwards and is shifted 4 units to the right compared to the graph of y = x^2. The vertex of G(x) is located at (4, 0) and it passes through the points (0, 16), (1, 9), (2, 4), (3, 1), and (4, 0).
To graph the function G(x) = x^2 + 4:
1. Start with the graph of y = x^2, which is a standard parabola that opens upward.
2. To obtain G(x), add 4 to the y-coordinate of each point on the graph of y = x^2. This will shift the graph upward by 4 units.
- For example, the point (0, 0) on the original graph becomes (0, 4) on the new graph.
3. Draw the new graph, keeping the same shape as the original graph but shifted upward by 4 units.
- The vertex of the parabola will still be at the origin (0, 0).
To graph the function G(x) = (x - 4)^2:
1. Start with the graph of y = x^2 as a reference again.
2. To obtain G(x), replace x with (x - 4) in the equation y = x^2. This will shift the graph 4 units to the right.
- For example, the point (0, 0) on the original graph becomes (4, 0) on the new graph.
3. Draw the new graph, keeping the same shape as the original graph but shifted 4 units to the right.
- The vertex of the parabola will now be at (4, 0).
By following these steps, you can graph both G(x) = x^2 + 4 and G(x) = (x - 4)^2.
To graph G(x) = x^2 + 4, you can start by understanding how this function is related to the graph of y = x^2.
1. Start with the graph of y = x^2:
- Plot points on a coordinate plane by selecting various x-values and finding the corresponding y-values by squaring the x-values.
- Connect the points smoothly to form a parabolic curve that opens upwards.
2. Now, let's analyze G(x) = x^2 + 4:
- The "+4" part of the equation indicates that the entire graph of G(x) will be shifted upward by 4 units compared to the basic parabola y = x^2.
- To achieve this, for each point on the graph of y = x^2, you will add 4 to its y-value to obtain the corresponding point on the graph of G(x).
3. Graph G(x) = x^2 + 4:
- Start with the points from the graph of y = x^2.
- Add 4 to the y-coordinate of each point from the graph of y = x^2.
- Plot the new points on the coordinate plane.
- Connect the new points smoothly to form the graph of G(x).
4. Additionally, if you want to graph G(x) = (x - 4)^2:
- This equation indicates that the entire graph of G(x) will be shifted 4 units to the right compared to the basic parabola y = x^2.
- To achieve this, for each point on the graph of y = x^2, you will shift the x-value 4 units to the right to obtain the corresponding point on the graph of G(x).
5. Graph G(x) = (x - 4)^2:
- Start with the points from the graph of y = x^2.
- Shift each x-coordinate 4 units to the right.
- Plot the new points on the coordinate plane.
- Connect the new points smoothly to form the graph of G(x).
By following these steps, you should be able to graph G(x) = x^2 + 4 and G(x) = (x - 4)^2 by understanding the transformations applied to the original graph of y = x^2.