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How do you graph

y = x + 2
y = 3x - 2

1 answer

To graph the equations \(y = x + 2\) and \(y = 3x - 2\), follow these steps:

Step 1: Graph \(y = x + 2\)

  1. Identify the Slope and Y-Intercept:

    • The equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
    • Here, the slope \(m = 1\) and the y-intercept \(b = 2\).

  2. Plot the Y-Intercept:

    • Start by plotting the point \((0, 2)\) on the graph.

  3. Use the Slope to Find Another Point:

    • From \((0, 2)\), use the slope to find another point. Since the slope is 1, move 1 unit up (rise) and 1 unit right (run) to get to \((1, 3)\).
    • Plot this point.

  4. Draw the Line:

    • Draw a straight line through the points \((0, 2)\) and \((1, 3)\). Extend the line in both directions.

Step 2: Graph \(y = 3x - 2\)

  1. Identify the Slope and Y-Intercept:

    • For this equation, the slope \(m = 3\) and the y-intercept \(b = -2\).

  2. Plot the Y-Intercept:

    • Plot the point \((0, -2)\) on the graph.

  3. Use the Slope to Find Another Point:

    • From \((0, -2)\), use the slope to find another point. Since the slope is 3, move 3 units up (rise) and 1 unit right (run) to get to \((1, 1)\).
    • Plot this point.

  4. Draw the Line:

    • Draw a straight line through the points \((0, -2)\) and \((1, 1)\). Extend the line in both directions.

Step 3: Find the Intersection Point

To find the intersection point of the two lines, set the equations equal to each other:

\[ x + 2 = 3x - 2 \]

  1. Rearranging gives:

\[ 2 + 2 = 3x - x \] \[ 4 = 2x \] \[ x = 2 \]

  1. Substitute \(x = 2\) back into either equation to find \(y\):

\[ y = 2 + 2 = 4 \]

So, the intersection point is \((2, 4)\).

Summary of the Graphing Process:

  • First Line: Start at \((0, 2)\), slope is 1, passes through \((1, 3)\).
  • Second Line: Start at \((0, -2)\), slope is 3, passes through \((1, 1)\).
  • Intersection Point: The lines intersect at \((2, 4)\).

Finally, make sure to label the lines and the intersection point on your graph for clarity!