1.2 Basic classes of functions

We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

Linear functions and slope

The easiest type of function to consider is a linear function    . Linear functions have the form $f(x)=ax+b,$ where $a$ and $b$ are constants. In [link] , we see examples of linear functions when $a$ is positive, negative, and zero. Note that if $a>0,$ the graph of the line rises as $x$ increases. In other words, $f(x)=ax+b$ is increasing on $\text{(−∞, ∞)}.$ If $a<0,$ the graph of the line falls as $x$ increases. In this case, $f(x)=ax+b$ is decreasing on $\text{(−∞, ∞)}.$ If $a=0,$ the line is horizontal.

As suggested by [link] , the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope    is the change in $y$ for each unit change in $x.$ The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in $y$ versus the change in $x.$ To do so, we choose any two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line and calculate $\frac{y_2-y_1}{x_2-x_1}.$ In [link] , we see this ratio is independent of the points chosen.

We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula $f\left(x\right)=ax+b.$ As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating $(y_2-y_1)\text{/}(x_2-x_1)$ for any points $(x_1,y_1)$ and $(x_2,y_2)$ on the line. Evaluating the function $f$ at $x=0,$ we see that $(0,b)$ is a point on this line. Evaluating this function at $x=1,$ we see that $(1,a+b)$ is also a point on this line. Therefore, the slope of this line is