Linear Functions Explorer
Explore linear functions through slope and y-intercept manipulation
- Understand how slope determines the steepness and direction of a line
- Identify y-intercept and x-intercept from the equation
- Convert between slope-intercept, point-slope, and general form
- Apply linear functions to model real-world scenarios
y = mx + b
m slope (gradient)
b y-intercept
A linear function has the form y = mx + b. The parameter m is the slope (gradient) of the line. It tells you how steep the line is and whether it goes up or down. A positive m means the line rises from left to right. A negative m means it falls. When m = 0, the line is horizontal. The parameter b is the y-intercept — the point where the line crosses the y-axis. Use the sliders to change m and b and watch how the green line moves compared to the blue dashed original.
Your Goal: Adjust slope and y-intercept to match the red target line, then solve real-world problems. Click New Challenge to start!
Real-World Connection: Linear functions model many everyday situations. A taxi charges a $3 flag-fall plus $2 per kilometre — the cost function C = 2d + 3 is linear with slope 2 and y-intercept 3. Phone plans, hourly wages, and conversion between Celsius and Fahrenheit all follow straight-line relationships.
Function Form
y = x
y = x
Parameters
The slope m tells you how steep the line is — rise over run. The y-intercept b is where the line crosses the vertical axis.
Function Type
Challenges
Interactive Graph
Watch the line pivot and shift as you change m and b. Two points are enough to define a unique line, and the equation y = mx + b captures it all.
Original
Your Line
Interpretation
These values are calculated from the equation. Parallel lines share the same slope; perpendicular lines have slopes that multiply to −1.
Slope: 1 (rising)
Y-intercept: (0, 0)
X-intercept: (0, 0)
Direction: Rising
Progress
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0Score
0Best Streak