Graphing Lines in Slope-Intercept Form

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Identify slope \(m\) and y-intercept \(b\) in \(y = mx + b\).
  • Graph lines by starting at the intercept and using slope.
  • Explain how slope and intercept affect the shape and position of a line.

Key Ideas

Slope-intercept form is the most graph-friendly way to write a linear equation:

\[ y = mx + b \]

  • \(m\) = slope (the “tilt” — rise over run)
  • \(b\) = y-intercept (where the line crosses the y-axis)

To graph a line:

  1. Plot the point \((0, b)\) on the y-axis.
  2. Use slope \(m = \frac{\text{rise}}{\text{run}}\) to move to a second point.
  3. Draw a straight line through both points.

A line showing the y-intercept and a rise/run triangle used to calculate slope.

Common Problem Types

1. Graphing using slope and intercept

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

  • \(m = 2 = \frac{2}{1}\)
  • \(b = -3\)

Plot \((0, -3)\), then go up 2, right 1.

2. Identify slope/intercept from a graph

  • Intercept = the point where the line touches the y-axis
  • Slope = compute rise/run using any two clear lattice points

3. Writing an equation from a description

Example: “Slope 4, passes through \((0, -1)\).”
Because \((0, -1)\) is the intercept:

\[ y = 4x - 1 \]

Strategies

  • Write slope as a fraction (even whole numbers like \(3\) → \(\frac{3}{1}\)).
  • Always start by plotting the y-intercept, not x-intercept.
  • Use a rise/run triangle to avoid sign mistakes.
  • If slope is negative, think “up and left” or “down and right.”
  • Use two points to draw a more accurate line (not just one).

Worked Examples

Example 1

Graph \(y = -\frac{1}{2}x + 4\).

  • \(b = 4\) → start at \((0, 4)\)
  • \(m = -\frac{1}{2}\) → down 1, right 2

Plot \((0, 4)\) and then \((2, 3)\), then draw the line.

Example 2

Find the slope-intercept equation from two points: \((1, 3)\) and \((5, 11)\).

Find slope:

\[ m = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2 \]

Plug into \(y = mx + b\) using point \((1, 3)\):

\[ 3 = 2(1) + b \Rightarrow b = 1 \]

Final equation:

\[ y = 2x + 1 \]

WarningCommon Mistakes
  • Using run/rise instead of rise/run.
  • Plotting the intercept on the x-axis by accident.
  • Forgetting that slope is directional and can be negative.
  • Dropping the sign on the y-intercept.

Practice Problems

  1. Graph \(y = 3x - 2\).
  2. Graph \(y = -x + 5\).
  3. Identify slope/intercept: \(y = \frac{1}{3}x - 4\).
  4. Write equation through \((0, -2)\) and \((3, 4)\).
  5. Write equation with slope \(-2\) and y-intercept \(7\).

3.
Slope = \(1/3\), intercept = \(-4\).

4.
\[ m = \frac{4 - (-2)}{3 - 0} = \frac{6}{3} = 2 \] So
\[ y = 2x - 2 \]

5.
\[ y = -2x + 7 \]

Summary

  • In \(y = mx + b\), slope tells how steep the line is, and \(b\) tells where it crosses the y-axis.
  • Graph by plotting the intercept and applying rise/run.
  • A line is fully determined by its slope and y-intercept.
  • Rewrite slope as a fraction to graph it more easily.
  • Start at the y-intercept every time.
  • Negative slope means the line goes down as you move right.
  • Use at least two points for accuracy.