1.4 Linear Functions

Introduction

Properties of linear functions

Let $y = f(x) = 7 - 3x$. Determine $AV_{[-3,-1]}$, $AV_{[2,5]}$, and $AV_{[4,10]}$ for the function $f$.

Find a formula for a linear function $f$ whose average rate of change is $m = -\frac{1}{4}$ and passes through the point $(-7,-5)$.

Solution. Using Equation and the facts that $m = -\frac{1}{4}$ and $f(-7) = -5$ (that is, $x_0 = -7$ and $f(x_0) = -5$), we have

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The line with slope $\frac{3}{7}$ that passes through $(-11, -17)$.

The line passing through the points $(-2,5)$ and $(3,-1)$.

The line passing through $(4,9)$ that is parallel to the line $2x - 3y = 5$.

Explain why the function $f$ given by the data in the following table appears to be linear and find a formula for $f(x)$.

$x$ | $1$ | $3$ | $4$ | $7$

$f(x)$ | $7$ | $3$ | $1$ | $-5$

Find a formula for the linear function shown in the following figure.

For the line given by $y = -3 - 2.5(x-5)$, determine its slope and a point that lies on the line.

Solution. This line is in point-slope form. Its slope is $m = -2.5$ and a point on the line is $(5,-3)$.

For the line given by $y = 6 + 0.25x$, determine its slope and a point that lies on the line.

Solution. This line is in slope-intercept form. Its slope is $m = 0.25$ and a point on the line is $(0,6)$, which is also the line's $y$-intercept.