Point Slope Graphing Notes
Algebra 1 • Coordinate Geometry
Name: Date: Period:
GRAPHING A LINE: POINT & SLOPE
Extra-Large Graph Paper Edition • Launchpad Point & Direction Vector
Grid Paper Practice Linear Functions
1. Launchpad Point
\((x_1, y_1)\)
Plot this point first. Do not start at the origin unless given \((0,0)\)!
2. Slope Ratio
\(m = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta y}{\Delta x}\)
Whole number: rewrite as \(\frac{m}{1}\). Negative slope: \(-\frac{2}{3} = \frac{-2}{3}\) (down 2, right 3).
3. Compass Guide
+ Rise: UP \(\uparrow\)
+ Run: RIGHT \(\rightarrow\)
- Rise: DOWN \(\downarrow\)
- Run: LEFT \(\leftarrow\)
Zero slope (\(m=0\)) = Horizontal line \(\leftrightarrow\)
1 Plot Start Point \((x_1, y_1)\)
2 Count Rise/Run along grid squares
3 Connect points & add arrows
Teacher Models
Graph Paper Demonstrations
Model 1: Positive Slope
Point: \((-3, -2)\) | Slope: \(m = \frac{3}{2}\)
Scale = 1
x y -4 -2 2 4 4 2 -2 -4 +3 +2
● Start Point: Plot \((-3, -2)\).
● Rise = +3: Count 3 squares UP.
● Run = +2: Count 2 squares RIGHT.
● New Point: Plot \((-1, 1)\). Repeat for \((1, 4)\)!
Model 2: Integer Slope
Point: \((1, 3)\) | Slope: \(m = -2\)
Scale = 1
x y -4 -2 2 4 4 2 -2 -4 -2 +1
● Start Point: Plot \((1, 3)\).
● Rewrite Over 1: \(-2 = \frac{-2}{1}\).
● Rise = -2: Count 2 squares DOWN.
● Run = +1: Count 1 square RIGHT.
Guided Practice
Try It on Graph Paper (Maximum Size Workspace)
Scale = 1 unit per square
Problem A Point: \((-2, 4)\) • Slope: \(m = -\frac{1}{2}\)
Start: ( , )
Rise:
Run:
x y -4 -2 2 4 4 2 -2 -4
Plot start point, count squares, and draw the line with arrows
Problem B Point: \((-1, -3)\) • Slope: \(m = \frac{4}{3}\)
Start: ( , )
Rise:
Run:
x y -4 -2 2 4 4 2 -2 -4
Plot start point, count squares, and draw the line with arrows
INDEPENDENT PRACTICE & GRAPH PAPER CHALLENGE
Count along the little grid squares. Clearly label the points and draw clean arrows at both ends.
Page 2 of 2
Part 1
Standard Scale (Each little square = 1 unit)
4 Practice Graphs
1 Problem 1
Point: \((2, 1)\)
Slope: \(m = -\frac{3}{4}\)
Rise:
Run:
x y -4 -2 2 4 4 2 -2 -4
2 Problem 2
Point: \((-3, 0)\)
Slope: \(m = \frac{2}{3}\)
Rise:
Run:
x y -4 -2 2 4 4 2 -2 -4
3 Problem 3
Point: \((1, -3)\)
Slope: \(m = 2\)
Rise:
Run:
x y -4 -2 2 4 4 2 -2 -4
4 Problem 4
Point: \((0, 3)\)
Slope: \(m = 0\) (Zero)
Type:
Line is:
x y -4 -2 2 4 4 2 -2 -4
Boss Level
The Scale Challenge: Watch the Axes!
⚠️ Don't just count squares! Look at the numbers!
5 Scale = 2 per box
Point: \((-4, -2)\) | Slope: \(m = \frac{3}{2}\)
Counts by 2s
x y -8 -4 4 8 8 4 -4 -8
1 box = 2 units!
Plot \((-4, -2)\):
2 boxes left, 1 box down.
Slope Trick:
\(\frac{3}{2} = \frac{6}{4}\)
Up 3 boxes
Right 2 boxes
6 Counts by 2s Every Other Box
Point: \((-3, 5)\) | Slope: \(m = -\frac{2}{3}\)
Every 2nd Box
x y -4 -2 2 4 4 2 -2 -4
Odd coordinates!
Where is \(x = -3\)?
Halfway between \(-2\) and \(-4\).
Count 2 squares down, 3 squares right. Connect carefully!
Mathematical Reflection: Why Scale Matters
In Problem 5, what mistake happens if you blindly count "3 squares up and 2 squares right" instead of checking the axis scale? Explain:
