Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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