Linear Functions: Slope & Y-Intercept

Description

This animation introduces linear functions of the form y = mx + b by visually demonstrating how the slope (m) and y-intercept (b) affect the graph. A line rotates as the slope changes from -2 to 2, then shifts vertically as the intercept changes from -3 to 3. A rise/run right-triangle indicator reinforces the geometric meaning of slope.

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Phases

#	Phase Name	Duration	Description
1	Introduction	5s	Display axes and the formula y = mx + b; draw the initial line at m=1, b=0
2	Slope Animation	15s	Animate m from -2 to 2, line rotates; show rise/run triangle updating live
3	Intercept Animation	12s	Animate b from -3 to 3, line shifts up and down; b label highlights
4	Summary	8s	Settle at m=1, b=1; label all parts; fade out with key takeaway text

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Layout

+-----------------------------------------------+
|  y = mx + b                   [formula top]   |
|                                                |
|        ^                                       |
|        |       /  <- line                      |
|        |      /                                |
|  ------+-----/-------->                        |
|        |    /  [rise/run triangle]             |
|        |   /                                   |
|                                                |
|  m = 1.00   b = 0.00         [value labels]   |
+-----------------------------------------------+

Area Descriptions

Area	Content	Notes
Center	Axes + plotted line	NumberPlane-style axes via Axes
Top center	MathTex formula y = mx + b	Updates color highlights as params change
Bottom left	m and b value readouts	Decimal displays driven by ValueTrackers
On graph	Rise/run right triangle	Drawn near x=1 to show slope visually

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Assets & Dependencies

• Fonts: LaTeX
• Colors: LINE_COLOR = YELLOW, SLOPE_COLOR = GREEN, INTERCEPT_COLOR = BLUE, TRIANGLE_COLOR = ORANGE, BG = #1a1a2e
• Manim version: ManimCE 0.19.1

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Notes

• ValueTracker for m and b; use always_redraw for the line and triangle
• Rise/run triangle anchored to a fixed x-window so it stays on screen during rotation
• Keep axes range x: [-5, 5], y: [-5, 5]