Q. Which Graph Represents \( y – 1 = 2(x – 2) \)?

Problem

Which graph represents the line given by the equation

\( y – 1 = 2(x – 2) \)

Step-by-step detailed solution

1. Expand and rewrite into slope-intercept form

   Start with the given point-slope form:

   \( y – 1 = 2(x – 2) \)

   First distribute the 2 on the right-hand side:

   \( y – 1 = 2x – 4 \)

   Next isolate \( y \) by adding 1 to both sides:

   \( y = 2x – 4 + 1 \)

   Combine like terms on the right-hand side:

   \( y = 2x – 3 \)

   This is the slope-intercept form \( y = mx + b \) with slope \( m = 2 \) and y-intercept \( b = -3 \).

2. Interpret slope and intercept

   The slope \( m = 2 \) means that for each increase of 1 in \( x \), \( y \) increases by 2. The y-intercept \( b = -3 \) means the line crosses the y-axis at the point \( (0,-3) \).

3. Generate points to plot the line

   Choose a few x-values, compute y using \( y = 2x – 3 \), and list the corresponding points:

   \(x\)	\(y = 2x – 3\)	Point
   0	\(2(0) – 3 = -3\)	\((0,-3)\)
   1	\(2(1) – 3 = -1\)	\((1,-1)\)
   2	\(2(2) – 3 = 1\)	\((2,1)\)
   \(\tfrac{3}{2}\)	\(2\left(\tfrac{3}{2}\right) – 3 = 0\)	\(\left(\tfrac{3}{2},0\right)\) (x-intercept)

   These points lie on the line and are sufficient to draw it accurately.

4. Describe the graph

   The graph is a straight line that:

   • passes through \( (0,-3) \) (y-intercept),
   • passes through \( (2,1) \) (the point given implicitly by the original point-slope form),
   • rises to the right because the slope is positive, and for every 1 unit increase in \( x \), \( y \) increases by 2, so the line is fairly steep upward.
5. Final identification

   The correct graph is the straight line with equation

   \( y = 2x – 3 \)

   So pick the graph that shows a line with slope 2 and y-intercept at \( (0,-3) \).