Linear Functions, Rate of Change, and Direct Variation

A line as a constant rate of change

What makes a relationship linear is that its rate of change is constant: every one-unit step in x changes y by the same amount, namely the slope m. That is exactly why the graph is a straight line. In a real model, the y-intercept b is the starting value and the slope m is the rate at which things grow or shrink.

A taxi charges $3 to start plus $2 per mile.
  cost y as a function of miles x:
     y = 2x + 3
  starting value b = 3   (the flat fee)
  rate m = 2             ($2 per mile)

  x=0:  y = 3      x=1:  y = 5      x=2:  y = 7
  every extra mile adds exactly $2  ->  constant rate

Lines as functions

A non-vertical line is a function: each input x gives exactly one output y. We often write a linear equation in function notation, f(x) = mx + b, so that f(3) means “the y-value when x is 3.” The slope-intercept form you already know is just this function written out.

f(x) = -3x + 10

  f(0) = -3(0) + 10 = 10     point (0, 10)
  f(2) = -3(2) + 10 = 4      point (2, 4)
  f(5) = -3(5) + 10 = -5     point (5, -5)

Each input x produces one output -> a function.
The outputs fall by 3 per step -> slope -3.

Direct variation: the cleanest line

Direct variation is the special case where the y-intercept is 0, so the line passes through the origin. We write it as y = kx, where k is the constant of proportionality — and it is exactly the slope. Doubling x doubles y; the ratio y/x stays fixed at k for every point on the line.

1. Recognize direct variation: the relationship can be written y = kx with no added constant.
2. Find k from one known pair (x, y) by dividing: k = y / x.
3. Write the rule y = kx, then use it to predict y for any new x.

y varies directly with x, and y = 18 when x = 6.

  k = y / x = 18 / 6 = 3
  rule:  y = 3x

Predict y when x = 10:
  y = 3(10) = 30

Check the ratio stays constant:
  18/6 = 3      30/10 = 3      same k