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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. € 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; € (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 € € vertical change = m ⋅ horizontal change, € € it follows that y − b = m( x − 0 ), or € y = mx + b. € we have justified the familiar formula for That is, the equation of a line. The quantities m and b are € 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 € into the equation y = mx + b, and substitute the value of m as well, leaving only b as unknown; € 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 € € determine the values of both parameters. €