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Chapter 7: Problem 60

Describe one similarity and one difference between the graphs of \(y^{2}=4 x\) and \((y-1)^{2}=4(x-1)\)

Short Answer

Expert verified
The similarity between the two graphs is that both are right-opening parabolas with the same spread. The difference between the two graphs lies in their positions in the Cartesian plane with the graph of \((y-1)^{2}=4(x-1)\) being shifted 1 unit to the right and 1 unit down from the position of the graph of \(y^{2}=4 x\)

Step by step solution

01

Understanding the Graphs

The equation \(y^{2}=4 x\) is a parabola that opens in the right direction. The equation \((y-1)^{2}=4(x-1)\) is also a parabola. By comparing these two equations, the second parabola can be considered as a shift of the first parabola. This shift is 1 unit downwards and 1 unit towards right.
02

Establishing the Similarity

The similarity between these two parabolas lies in their shapes and orientations. Both are right-opening parabolas, indicating that both have the same direction of opening. Furthermore, both parabolas have the same width, given by the coefficient 4 of x in both equations, indicating that the spread of both parabolas is the same.
03

Finding the Difference

The main difference between these two graphs is their positions in the Cartesian plane. The second graph is a translation or shift of the first graph. It is shifted 1 unit to the right and 1 unit down from the position of the first graph. This is reflected in the h and k values of the general parabula equation \((y - h)^2 = 4p (x - k)\), indicating shifts along the x and y axes respectively

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