Problem 72 - 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 graphs of \(y^{2}=4 x\) and \((y-1)^{2}=4(x-1)\) is that both are parabolas opening towards the positive x-axis and have the same width. The difference is their location on the coordinate plane. The first parabola has the vertex at the origin (0,0) while the second parabola is shifted 1 unit up and 1 unit to the right, having the vertex at (1,1).

Step 1: Graph Analysis

Observe the two equations. The first equation, \(y^{2}=4 x\), represents a standard parabola opening to the right with a vertex at the origin (0,0). The second equation, \((y-1)^{2}=4(x-1)\), represents a similar parabola but translated one unit to the right and one unit up. This makes the vertex of the parabola (1,1).

Step 2: Finding the Similarity

The similarity between the two is their shape and orientation. Both equations represent parabolas that open towards the right, i.e., positive x-axis. The '4' in both equations determines how 'wide' or 'narrow' the parabola is. Since the value is the same in both equations, the parabolas have the same 'width'.

Step 3: Finding the Difference

The difference between the two parabolas lies in their locations on the coordinate plane. The vertex of the parabola given by \(y^{2}=4 x\) is at the origin, (0,0). The vertex of \((y-1)^{2}=4(x-1)\) is at (1,1) which implies that the parabola is shifted 1 unit up and 1 unit to the right.