2.1 Linear Functions

A linear function is a function whose graph is a straight line.

Thus, if f is a linear function, then

f ( x ) = a x + b

for constants a and b, where a is the slope of the line and (0, b) is the y-intercept of the line.

From the fact that two points determine a line it follows that the equation of a linear function can be determined from knowing the output values corresponding to any two input values.

For example, suppose that we know that f is a linear function and we know that f (1) = - 2 and that f (4) = 3.

Then it follows that the points (1,- 2) and (4,3) lie on the graph of the function.

We compute the slope of the line by dividing the difference of the outputs by the difference of the inputs, getting the result 5 / 3.

Thus, a = 5 / 3.

To discover the value of b, we note that

f ( x ) = (5 / 3) x + b

Thus, when x = 1

f ( 1 ) = (5 / 3) + b

Thus

- 2 = (5 / 3) + b

From which it follows that b = - 11 / 3 .

Thus, the equation of the function is

f ( x ) = (5 / 3) x - (11 / 3)

Exercise 2.1.1

The function T which converts degrees Fahrenheit to degrees Celsius is a linear function. Given that T( 98.6 ) = 37 and T( 32 ) = 0, find the equation of the function.

Solution

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