How can the graph of f(x)=1/2(x+10)^2-2 be obtained from the graph of y=x^2
They are both parabolas that "open up" in the +y direction, but the vertices are in different places. y = x^2 has a vertex (minimum) at (0,0), and the other function has a vertex at (-10,-2)
The y = x^2 function opens up more slowly, which means it is narrower at a given distance above the vertex.
To obtain the graph of the function f(x) = 1/2(x+10)^2 - 2 from the graph of y = x^2, you can follow these steps:
1. Start with the graph of y = x^2. This is a standard parabolic graph that opens upward, passing through the origin.
2. To obtain f(x), you need to perform two transformations on the graph of y = x^2: translation and vertical stretching/compression.
3. The first transformation is a translation, which involves shifting the graph horizontally and vertically. In this case, f(x) = 1/2(x+10)^2 - 2, suggests that the parabola is shifted 10 units to the left (horizontal translation) and 2 units down (vertical translation).
4. To shift the graph 10 units to the left, you would move each point on the graph 10 units to the left. For example, the point (1, 1) on the original parabola would move to (-9, 1) on the new graph.
5. To shift the graph 2 units down, you would move each point on the graph 2 units down. For example, the point (1, 1) on the original parabola would move to (1, -1) on the new graph.
6. The second transformation is vertical stretching/compression. The coefficient 1/2 in front of (x+10)^2 indicates that the graph is vertically compressed by a factor of 1/2. This means that the y-values of each point on the graph are halved.
7. Combining both transformations, you would shift the original graph of y = x^2 10 units to the left, 2 units down, and then vertically compressed by a factor of 1/2.
8. Repeat steps 4 and 5 for a few key points on the original graph, such as the vertex (0, 0), and plot the corresponding points on the new graph following the transformations.
9. Connect all the points on the new graph to form the revised parabola.
By following these steps, you can obtain the graph of f(x) = 1/2(x+10)^2 - 2 from the graph of y = x^2.