Problem 80 - Algebra and Trigonometry

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 parabolas \(y^{2}=4x\) and \((y-1)^{2}=4(x-1)\) are similar in their shape and orientation - both open to the right. The difference is in their location: the former is at the origin, while the latter is shifted one unit to the right and one unit upwards.

Step 1: Identify the Graphs

First, identify the shapes of the two graphs. Both equations represent a parabola, as they are quadratic equations. The first equation, \(y^{2}=4x\), opens to the right because the positive \(y^{2}\) term is equal to a positive constant times \(x\). The second equation, \((y-1)^{2}=4(x-1)\), will have the same orientation and shape because it is a transformation of the first.

Step 2: Scrutinize the Transformations

The graph \((y-1)^{2}=4(x-1)\) is a result of translation of the graph \(y^{2}=4x\). The '1' subtracted from both \(y\) and \(x\) values in thesecond equation indicates a shift. It shifts the entire graph of \(y^{2}=4x\) one unit to the right and one unit upwards.

Step 3: Draw the Similarities and Differences

Both graphs represent parabolas that open to the right. The similarity lies in their shape and orientation. However, they are different in terms of position. The graph of \(y^{2}=4x\) is located at the origin, while the graph of \((y-1)^{2}=4(x-1)\) is shifted one unit to the right and one unit upwards from the original position.