Download Linear functions: graphs and formulas A linear function is one whose

Section 3.2
1
Linear functions: graphs and formulas
A linear function is one whose graph is a straight
line. Consequently, it has the same slope between
any pair of points; that is, there is a constant rate
of change between any pair of inputs.
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Suppose y = f ( x ) is a linear function. Then its
initial output value is the output for input x = 0,
which is often denoted in function notation by
f (0 ) = b . Graphically, this is the y-intercept of the
€ that is, the line must pass through the point
line;
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(0,b ). If ( x, y ) represents any other point on the
line, then the slope between (0,b ) and ( x, y ) must
equal the slope m of the entire line. Since
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vertical change = m ⋅ horizontal change,
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it follows that y − b = m( x − 0 ), or
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y = mx + b.
€ we have justified the familiar formula for
That is,
the equation of a line. The quantities m and b are
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called the parameters
or coefficients of the
equation (like variables they don’t have specified
values, but unlike variables, they do not vary!). It
follows that
Section 3.2
2
y = rate of change ⋅ x + initial output.
The x-intercept of the line corresponds to where the
output y = 0; we can determine it from the formula
€by setting y = 0 and solving for x. The simple form
of the equation makes this a straightforward task.
€Notice that the equation of the line is determined
by€its two parameters, m and b. This corresponds
precisely to the geometric law that two points
determine the line.
This phenomenon works more generally. To find
the equation of a linear function, we need only two
bits of independent information:
• slope m and initial value b: this directly gives
y = mx + b.
• slope m and one point ( p,q ) on the line:
substitute the coordinates ( p,q ) for the variables
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into the equation y = mx + b, and substitute the
value of m as well, leaving only b as unknown;
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solve for b to determine both parameters.
€ (s,t ): calculate m first as
• two points, ( p,q ) and
the slope €
between the two points; then, as in the
previous case, substitute m and the coordinates of
either point into y = mx + b, then solve for b to
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determine
the values of both parameters.
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