Working with Multiple Plots

Compare functions, find intersections, and analyze relationships

Why Plot Multiple Functions?

Plotting multiple functions on the same graph is one of the most powerful features of mathematical visualization. It allows you to:

Compare behavior: See how different functions grow, decay, or oscillate
Find intersections: Identify where functions meet or cross
Analyze relationships: Understand how functions relate to each other
Solve equations: Graphically solve f(x) = g(x) problems
Visualize transformations: Compare original functions with their modifications

Adding Multiple Plots in FooPlot

Method 1: Using "Add Plot" Button

Enter your first function (e.g., x^2)
Click "Plot" to display it
Click the "Add Plot" button
Enter your second function (e.g., 2*x + 1)
Click "Plot" again
Both functions now appear on the same graph

Method 2: Managing Multiple Plots

Once you have multiple plots, you can:

Toggle visibility: Click the eye icon to show/hide individual plots
Change colors: Each plot gets a different color automatically
Remove plots: Click the delete button to remove unwanted plots
Reorder plots: Manage which plots appear on top

Common Multiple Plot Scenarios

1. Comparing Polynomial Degrees

Linear vs Quadratic vs Cubic

Plot 1: x (linear) Plot 2: x^2 (quadratic) Plot 3: x^3 (cubic)

See how the growth rate changes with polynomial degree.

Different Quadratics

Plot 1: x^2 Plot 2: 2*x^2 Plot 3: x^2 + 3 Plot 4: (x-2)^2

Compare the effects of scaling and shifting.

2. Trigonometric Comparisons

Sine vs Cosine

Plot 1: sin(x) Plot 2: cos(x)

See the phase relationship between sine and cosine.

Different Frequencies

Plot 1: sin(x) Plot 2: sin(2*x) Plot 3: sin(3*x)

Compare how frequency affects wave patterns.

3. Exponential vs Logarithmic

Plot 1: exp(x) Plot 2: ln(x) Plot 3: x (reference line)

These functions are inverses of each other - see how they're symmetric about the line y = x.

Finding Intersections

Graphical Method

When you plot multiple functions, intersection points appear where the curves cross. To find intersections:

Plot both functions
Look for crossing points
Use zoom controls to get more precise coordinates
Adjust axis ranges for better view of intersection regions

Example: Find where x² = 2x + 3
Plot 1: x^2
Plot 2: 2*x + 3
The intersections occur at approximately x = -1 and x = 3.

Common Intersection Problems

Quadratic = Linear: Usually 0, 1, or 2 intersection points
Trigonometric intersections: Often multiple periodic intersections
Exponential = Polynomial: Typically 1-2 intersection points
Circle intersections: 0, 1, or 2 points depending on relationship

Analyzing Function Relationships

Growth Rate Comparisons

Fast Growing Functions: Plot 1: exp(x) (exponential - fastest) Plot 2: x^3 (cubic polynomial) Plot 3: x^2 (quadratic) Plot 4: x (linear - slowest)

Oscillation Comparisons

Damped vs Undamped Oscillations: Plot 1: sin(x) (constant amplitude) Plot 2: exp(-x/5)*sin(x) (damped oscillation) Plot 3: x*sin(x) (growing amplitude)

Function Transformations

Comparing Original with Transformations

Vertical Shifts

Plot 1: x^2 Plot 2: x^2 + 3 Plot 3: x^2 - 2

Horizontal Shifts

Plot 1: x^2 Plot 2: (x-2)^2 Plot 3: (x+1)^2

Scaling

Plot 1: x^2 Plot 2: 2*x^2 Plot 3: 0.5*x^2

Reflections

Plot 1: x^2 Plot 2: -x^2 Plot 3: (-x)^2

Practical Applications

Economics and Business

Revenue vs Cost Analysis: Plot 1: 50*x - 0.1*x^2 (Revenue function) Plot 2: 20*x + 100 (Cost function) Plot 3: 30*x - 0.1*x^2 - 100 (Profit = Revenue - Cost)

Physics Applications

Motion Analysis: Plot 1: 5*t (constant velocity) Plot 2: 0.5*9.8*t^2 (free fall distance) Plot 3: 20*t - 0.5*9.8*t^2 (projectile with initial velocity)

Signal Processing

Frequency Analysis: Plot 1: sin(x) + sin(3*x) (composite signal) Plot 2: sin(x) (fundamental frequency) Plot 3: sin(3*x) (harmonic)

Advanced Multiple Plot Techniques

Piecewise Function Comparison

Compare different pieces of piecewise functions:

Compare parts of |x|: Plot 1: x (for x ≥ 0) Plot 2: -x (for x ≤ 0) Range: -5 to 5

Approximation Analysis

Compare functions with their approximations:

Taylor Series Approximation of sin(x): Plot 1: sin(x) (actual function) Plot 2: x (1st order approximation) Plot 3: x - x^3/6 (3rd order approximation) Plot 4: x - x^3/6 + x^5/120 (5th order approximation)

Tips for Effective Multiple Plotting

🎯 Pro Tips:

Use appropriate ranges: Choose x and y ranges that show all important features
Color coding: Use the automatic color coding to distinguish functions
Legend awareness: Keep track of which color represents which function
Zoom strategically: Zoom in on intersection points for better analysis
Start simple: Begin with 2-3 functions before adding more

Common Mistakes to Avoid

⚠️ Avoid These Pitfalls:

Too many plots: More than 5-6 functions can become confusing
Similar colors: Make sure functions are visually distinct
Wrong ranges: Important features might be outside the viewing window
Scale issues: Functions with very different scales may not show well together

Interactive Features

Show/Hide Functionality

Use the visibility toggle to:

Focus on specific function pairs
Reduce visual clutter
Create step-by-step reveals
Compare subsets of your functions

Dynamic Range Adjustment

Adjust ranges to:

Show global behavior (wide range)
Focus on local features (narrow range)
Center on intersection points
Compare growth rates at different scales

Exercise Challenges

Polynomial Comparison: Plot x, x², x³, and x⁴ on the same graph. Observe their behavior for x ∈ [-2, 2]
Trigonometric Family: Plot sin(x), cos(x), and tan(x). What ranges work best?
Exponential Growth: Compare 2ˣ, 3ˣ, and eˣ. Which grows fastest?
Intersection Hunt: Find all intersections between x² and 2ˣ
Transformation Study: Plot f(x) = x² and its transformations: f(x+1), f(x-1), 2f(x), f(2x)
Inverse Functions: Plot f(x) = x³ and g(x) = ∛x along with y = x

Conclusion

Multiple plotting is a powerful tool for mathematical analysis and understanding. By comparing functions visually, you can gain insights that would be difficult to achieve through algebra alone. Practice with different combinations of functions, and don't hesitate to experiment with various viewing ranges and visibility settings to get the clearest picture of function relationships.

Remember: the goal is insight, not just pretty pictures. Always ask yourself what the multiple plots are telling you about the mathematical relationships involved.