Interactive Visualizations

Number line moves

Change a starting value and a step to see addition and subtraction as movement on a number line.

Start value 2 Step size 4

Update 2 + 4 = 6

Distance from zero 6

Truth table explorer

Switch the proposition values and connective to see how the truth table changes row by row.

Connective and

Inputs

Statement P and Q

Result false

P	Q	Value	Row
F	F	F	1
F	T	F	2
T	F	F	3
T	T	T	4

The highlighted row shows the current input combination and output.

Triangle angle and area

Adjust an included angle and the two side lengths to see how triangle area and the third side respond.

Side a 7 Side b 8 Included angle (deg) 62

Area 0

Third side 0

Linear equation explorer

Change slope and intercept to see how y = mx + b shifts and tilts on the coordinate plane.

Slope m 1.4 Intercept b -1

Equation y = 1.4x - 1.0

Intercepts x = 0.7, y = -1.0

Function family transformer

Choose a function family and shift, stretch, or lift it to see how the graph changes.

Family quadratic Vertical scale a 1.2 Horizontal shift h 1 Vertical shift k 0

Current form y = 1.2 f(x - 1.0) + 0.0

Domain note All real numbers

Binomial distribution

Adjust the success probability and trial count to see how the distribution of outcomes changes.

Success probability p 0.5 Trials n 6

Expected value 3.0

Variance 1.5

Dot plot and summary

Move five sample points to see how the mean, median, and spread respond to the data.

Data point 1 18 Data point 2 32 Data point 3 54 Data point 4 68 Data point 5 86

Mean 51.6

Median 54

Range 68

Unit circle and wave

Move the angle around the unit circle to watch sine and cosine update together.

Angle 30

sin(theta) 0.500

cos(theta) 0.866

tan(theta) 0.577

Approaching a limit

Move the removable discontinuity to see how the function approaches the same value from both sides.

Hole position a 1 Probe offset 0.6

Limit 2.0

Left/right values approach the same height

Tangent line slope

Move the tangent point to see the instantaneous slope on a cubic curve.

Point x0 0.8

Slope 0.92

Derivative f'(x0)

Area under a curve

Move the bounds to see how the accumulated area changes between two x-values.

Left bound a -1.2 Right bound b 1.8

Area 0.00

Riemann estimate 0.00

Geometric partial sums

Adjust the ratio and term count to see how a geometric series approaches its limit.

Ratio r 0.5 Terms n 6

Partial sum 1.969

Limit 2.000

Vector addition

Change the x and y components of two vectors and watch the resultant update in real time.

Vector A x 5 Vector A y 3 Vector B x -2 Vector B y 4

Resultant (3, 7)

Magnitude 7.62

Linear transformation

Change the entries of a 2x2 matrix to see how it stretches and shears a grid.

a 1.25 b 0.5 c -0.25 d 1

Determinant 1.375

Trace 2.25

Eigenvector directions

Use a symmetric matrix so the real eigenvectors stay visible as the transformation changes.

a 1.5 b 0.75 d 0.25

Eigenvalue 1 1.87

Eigenvalue 2 -0.12

Counting paths in a tree

Adjust the depth and branching factor to see how recursion and the product rule grow the number of outcomes.

Depth 4 Branches 2

Leaves 16

Count 2^4

Growth model

Adjust the initial value and growth rate to see how a simple model changes over time.

Initial value 5 Growth rate 0.08

Value at t = 5 7.35

Trend growth

Slope field and solution

Move the initial value to see how the solution curve follows the same slope field.

Equilibrium level 1 Initial value -1

Solution y(t) = 1 + Ce^-t

Target level 1.0

Damped oscillator

Tweak damping and frequency to see how the oscillation fades and tightens over time.

Damping 0.18 Frequency 2

Period 3.14

Decay slow

Phase portrait

Adjust the linear system to see trajectories spiral, settle, or diverge in the xy-plane.

Growth term 0.2 Rotation term 1

Stability spiral source

Initial point (1, -0.5)

Proof strategy map

Choose a proof strategy and watch the same claim reorganize into a different reasoning path.

Strategy direct

Claim If n is even, then n^2 is even

First step Write n = 2k

The highlighted path shows the reasoning order that matches the chosen strategy.

Free-body diagram explorer

Switch between a simply supported beam and a block on an incline to see which external forces belong on the isolated-body sketch.

Beam beam

Point load P 12 kN Load position x 2.5 m

Weight W 10 kN Incline angle 25 deg Friction coefficient mu 0.35

Forces Ay = 7.2 kN, By = 4.8 kN

Check Moment check: 24.0 = 24.0 kN m

The drawing shows external forces only. Internal forces are intentionally omitted.

T-s diagram explorer

Switch between a process path and a closed cycle to see how temperature and entropy move together on a T-s plane.

Process path process

Path type isobaric Start temperature T1 420 K Entropy span delta s 0.9 Temperature rise delta T 120 K

Hot temperature Th 650 K Cold temperature Tl 320 K Entropy width delta s 0.9

State detail Start T1 = 420 K

State detail Path type = isobaric

Heat proxy End state: T2 = 540 K, s2 = 1.55

Net result q_rev proxy = 108.0 kJ/kg

The filled region gives a simple heat-transfer proxy on the T-s plane.