Instructor Reference
Key Takeaway: Negative slopes always fall from left to right (\(\searrow\)) and pass through the origin \((0,0)\).
Form: \(y = mx\) where \(m < 0\)
3 Solution: \(y = -2x\)
Slope: \(m = \frac{-2}{1}\)
Slope (\(m\))
\(-2\) or \(\frac{-2}{1}\)
Rise / Fall
\(-2\) (Down 2)
Run (\(\Delta x\))
\(+1\) (Right 1)
Completed Table:
| \(x\) | Work: \(y = -2x\) | \(y\) | Point \((x, y)\) |
|---|---|---|---|
| -2 | \(-2(-2) = 4\) | 4 | (-2, 4) |
| -1 | \(-2(-1) = 2\) | 2 | (-1, 2) |
| 0 | \(-2(0) = 0\) | 0 | (0, 0) |
| 1 | \(-2(1) = -2\) | -2 | (1, -2) |
| 2 | \(-2(2) = -4\) | -4 | (2, -4) |
From left to right, this line is: Falling (\(\searrow\))
Line: \(y = -2x\)
-6 -4 -2 2 4 6 6 4 2 -2 -4 -6 x y
4 Solution: \(y = -\frac{3}{4}x\)
Slope: \(m = \frac{-3}{4}\)
Slope (\(m\))
\(-\frac{3}{4}\)
Rise / Fall
\(-3\) (Down 3)
Run (\(\Delta x\))
\(+4\) (Right 4)
Completed Table:
| \(x\) | Work: \(y = -\frac{3}{4}x\) | \(y\) | Point \((x, y)\) |
|---|---|---|---|
| -4 | \(-\frac{3}{4}(-4) = 3\) | 3 | (-4, 3) |
| 0 | \(-\frac{3}{4}(0) = 0\) | 0 | (0, 0) |
| 4 | \(-\frac{3}{4}(4) = -3\) | -3 | (4, -3) |
Starting at \((0,0)\), to find the next point go: Down 3, Right 4
Line: \(y = -\frac{3}{4}x\)
-6 -4 -2 2 4 6 6 4 2 -2 -4 -6 x y
Slope Pattern Synthesis: Answer Guide
1. Direction Rule: When \(m > 0\), the line slants upward (\(\nearrow\)). When \(m < 0\), it slants downward (\(\searrow\)).
2. The Origin: Why do ALL graphs of \(y = mx\) pass through \((0,0)\)?
When \(x = 0\), \(y = m(0) = 0\). Since there is no constant \(b\) added, \((0,0)\) is always a true solution.
Teacher Answer Key Page 3 Solutions
Instructor Reference
Key Takeaway: Rise over Run uses slope triangles directly from \((0,0)\). Dashed lines illustrate the \(\frac{\text{Rise}}{\text{Run}}\) steps.
Technique: Point Chaining
5 Solution: \(y = \frac{3}{2}x\)
Rise = +3, Run = +2
Vertical Rise (\(\Delta y\)) Move 3 Up
Horizontal Run (\(\Delta x\)) Move 2 Right
Completed Point Chain:
Point 1 (Start): Origin \((0, 0)\)
Point 2: From \((0,0)\), count Up 3, Right 2 \(\to\) (2, 3)
Point 3: From Point 2, count Up 3, Right 2 \(\to\) (4, 6)
Point 4 (Reverse): From \((0,0)\), count Down 3, Left 2 \(\to\) (-2, -3)
Slope triangle connects \((0,0) \to (2,0) \to (2,3)\).
Line & Slope Triangle: \(y = \frac{3}{2}x\)
+2 +3 -6 -4 -2 2 4 6 6 4 2 -2 -4 -6 x y
6 Solution: \(y = -\frac{1}{3}x\)
Fall = 1, Run = 3
Vertical Fall (\(\Delta y\)) Move 1 Down
Horizontal Run (\(\Delta x\)) Move 3 Right
Completed Point Chain:
Point 1 (Start): Origin \((0, 0)\)
Point 2: From \((0,0)\), count Down 1, Right 3 \(\to\) (3, -1)
Point 3: From Point 2, count Down 1, Right 3 \(\to\) (6, -2)
Point 4 (Reverse): From \((0,0)\), count Up 1, Left 3 \(\to\) (-3, 1)
Slope triangle connects \((0,0) \to (3,0) \to (3,-1)\).
Line & Slope Triangle: \(y = -\frac{1}{3}x\)
+3 -1 -6 -4 -2 2 4 6 6 4 2 -2 -4 -6 x y
Quick Check Answer Key
A. If \(m = \frac{5}{1}\), count UP 5 and RIGHT 1.
B. If \(m = -\frac{3}{5}\), count DOWN 3 and RIGHT 5.
