Linear Functions and Slope

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

As suggested by (Figure) , 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 [latex]y[/latex] for each unit change in [latex]x[/latex]. 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 [latex]y[/latex] versus the change in [latex]x[/latex]. To do so, we choose any two points [latex](x_1,y_1)[/latex] and [latex](x_2,y_2)[/latex] on the line and calculate [latex]\frac{y_2-y_1}{x_2-x_1}[/latex]. In (Figure) , 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 [latex]f(x)=ax+b[/latex]. 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 [latex](y_2-y_1)/(x_2-x_1)[/latex] for any points [latex](x_1,y_1)[/latex] and [latex](x_2,y_2)[/latex] on the line. Evaluating the function [latex]f[/latex] at [latex]x=0[/latex], we see that [latex](0,b)[/latex] is a point on this line. Evaluating this function at [latex]x=1[/latex], we see that [latex](1,a+b)[/latex] is also a point on this line. Therefore, the slope of this line is

We have shown that the coefficient [latex]a[/latex] is the slope of the line. We can conclude that the formula [latex]f(x)=ax+b[/latex] describes a line with slope [latex]a[/latex]. Furthermore, because this line intersects the [latex]y[/latex]-axis at the point [latex](0,b)[/latex], we see that the [latex]y[/latex]-intercept for this linear function is [latex](0,b)[/latex]. We conclude that the formula [latex]f(x)=ax+b[/latex] tells us the slope, [latex]a[/latex], and the [latex]y[/latex]-intercept, [latex](0,b)[/latex], for this line. Since we often use the symbol [latex]m[/latex] to denote the slope of a line, we can write

to denote the slope-intercept form of a linear function.

Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point [latex](x_1,y_1)[/latex] and the slope of the line is [latex]m[/latex]. Since any other point [latex](x,f(x))[/latex] on the graph of [latex]f[/latex] must satisfy the equation

this linear function can be expressed by writing

We call this equation the point-slope equation for that linear function.

Since every nonvertical line is the graph of a linear function, the points on a nonvertical line can be described using the slope-intercept or point-slope equations. However, a vertical line does not represent the graph of a function and cannot be expressed in either of these forms. Instead, a vertical line is described by the equation [latex]x=k[/latex] for some constant [latex]k[/latex]. Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation

where [latex]a,b[/latex] are both not zero, to denote the standard form of a line.

Polynomials

A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form

for some integer [latex]n\ge 0[/latex] and constants [latex]a_n, \, a_{n-1}, \cdots, a_0[/latex], where [latex]a_n\ne 0[/latex]. In the case when [latex]n=0[/latex], we allow for [latex]a_0=0[/latex]; if [latex]a_0=0[/latex], the function [latex]f(x)=0[/latex] is called the zero function . The value [latex]n[/latex] is called the degree of the polynomial; the constant [latex]a_n[/latex] is called the leading coefficient . A linear function of the form [latex]f(x)=mx+b[/latex] is a polynomial of degree 1 if [latex]m\ne 0[/latex] and degree 0 if [latex]m=0[/latex]. A polynomial of degree 0 is also called a constant function . A polynomial function of degree 2 is called a quadratic function . In particular, a quadratic function has the form [latex]f(x)=ax^2+bx+c[/latex], where [latex]a\ne 0[/latex]. A polynomial function of degree 3 is called a cubic function.

Algebraic Functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials. For example,

are rational functions. A root function is a power function of the form [latex]f(x)=x^{1/n}[/latex], where [latex]n[/latex] is a positive integer greater than one. For example, [latex]f(x)=x^{1/2}=\sqrt{x}[/latex] is the square-root function and [latex]g(x)=x^{1/3}=\sqrt[3]{x}[/latex] is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, [latex]f(x)=\sqrt{4-x^2}[/latex] is an algebraic function.

The root functions [latex]f(x)=x^{1/n}[/latex] have defining characteristics depending on whether [latex]n[/latex] is odd or even. For all even integers [latex]n \ge 2[/latex], the domain of [latex]f(x)=x^{1/n}[/latex] is the interval [latex][0,\infty)[/latex]. For all odd integers [latex]n \ge 1[/latex], the domain of [latex]f(x)=x^{1/n}[/latex] is the set of all real numbers. Since [latex]x^{1/n}=(-x)^{1/n}[/latex] for odd integers [latex]n, \, f(x)=x^{1/n}[/latex] is an odd function if [latex]n[/latex] is odd. See the graphs of root functions for different values of [latex]n[/latex] in (Figure) .

Transcendental Functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are [latex]\sin x,\, \cos x, \, \tan x, \, \cot x,\, \sec x[/latex], and [latex]\csc x[/latex]. (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form [latex]f(x)=b^x[/latex], where the base [latex]b >0, \, b \ne 1[/latex]. A logarithmic function is a function of the form [latex]f(x)=\log_b(x)[/latex] for some constant [latex]b >0, \, b \ne 1[/latex], where [latex]\log_b(x)=y[/latex] if and only if [latex]b^y=x[/latex]. (We also discuss exponential and logarithmic functions later in the chapter.)

Piecewise-Defined Functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function . The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of [latex]x[/latex]:

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for [latex]x < a[/latex] and [latex]x >a[/latex], we need to pay special attention to what happens at [latex]x=a[/latex] when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at [latex]x=a[/latex]. We examine this in the next example.

Graphing a Piecewise-Defined Function

Sketch a graph of the following piecewise-defined function:

[latex]f(x)=\begin{cases} x+3, & x < 1 \\ (x-2)^2 & x \ge 1 \end{cases}[/latex]

Solution

Graph the linear function [latex]y=x+3[/latex] on the interval [latex](-\infty,1)[/latex] and graph the quadratic function [latex]y=(x-2)^2[/latex] on the interval [latex][1,\infty )[/latex]. Since the value of the function at [latex]x=1[/latex] is given by the formula [latex]f(x)=(x-2)^2[/latex], we see that [latex]f(1)=1[/latex]. To indicate this on the graph, we draw a closed circle at the point [latex](1,1)[/latex]. The value of the function is given by [latex]f(x)=x+2[/latex] for all [latex]x < 1[/latex], but not at [latex]x=1[/latex]. To indicate this on the graph, we draw an open circle at [latex](1,4)[/latex].

This piecewise-defined function is linear for [latex]x < 1[/latex] and quadratic for [latex]x \ge 1[/latex].

Transformations of Functions

We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function [latex]y=x^2[/latex] to get the function [latex]f(x)=(x-2)^2[/latex]. This subtraction represents a shift of the function [latex]y=x^2[/latex] two units to the right. A shift, horizontally or vertically, is a type of transformation of a function . Other transformations include horizontal and vertical scalings, and reflections about the axes.

A vertical shift of a function occurs if we add or subtract the same constant to each output [latex]y[/latex]. For [latex]c >0[/latex], the graph of [latex]f(x)+c[/latex] is a shift of the graph of [latex]f(x)[/latex] up [latex]c[/latex] units, whereas the graph of [latex]f(x)-c[/latex] is a shift of the graph of [latex]f(x)[/latex] down [latex]c[/latex] units. For example, the graph of the function [latex]f(x)=x^3+4[/latex] is the graph of [latex]y=x^3[/latex] shifted up 4 units; the graph of the function [latex]f(x)=x^3-4[/latex] is the graph of [latex]y=x^3[/latex] shifted down 4 units ( (Figure) ).

A horizontal shift of a function occurs if we add or subtract the same constant to each input [latex]x[/latex]. For [latex]c >0[/latex], the graph of [latex]f(x+c)[/latex] is a shift of the graph of [latex]f(x)[/latex] to the left [latex]c[/latex] units; the graph of [latex]f(x-c)[/latex] is a shift of the graph of [latex]f(x)[/latex] to the right [latex]c[/latex] units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example.

Consider the function [latex]f(x)=|x+3|[/latex] and evaluate this function at [latex]x-3.[/latex] Since [latex]f(x-3)=|x|[/latex] and [latex]x-3 < x[/latex], the graph of [latex]f(x)=|x+3|[/latex] is the graph of [latex]y=|x|[/latex] shifted left 3 units. Similarly, the graph of [latex]f(x)=|x-3|[/latex] is the graph of [latex]y=|x|[/latex] shifted right 3 units ( (Figure) ).

A vertical scaling of a graph occurs if we multiply all outputs [latex]y[/latex] of a function by the same positive constant. For [latex]c >0[/latex], the graph of the function [latex]cf(x)[/latex] is the graph of [latex]f(x)[/latex] scaled vertically by a factor of [latex]c[/latex]. If [latex]c >1[/latex], the values of the outputs for the function [latex]cf(x)[/latex] are larger than the values of the outputs for the function [latex]f(x)[/latex]; therefore, the graph has been stretched vertically. If [latex]0 < c < 1[/latex], then the outputs of the function [latex]cf(x)[/latex] are smaller, so the graph has been compressed. For example, the graph of the function [latex]f(x)=3x^2[/latex] is the graph of [latex]y=x^2[/latex] stretched vertically by a factor of 3, whereas the graph of [latex]f(x)=x^2/3[/latex] is the graph of [latex]y=x^2[/latex] compressed vertically by a factor of 3 ( (Figure) ).

The horizontal scaling of a function occurs if we multiply the inputs [latex]x[/latex] by the same positive constant. For [latex]c >0[/latex], the graph of the function [latex]f(cx)[/latex] is the graph of [latex]f(x)[/latex] scaled horizontally by a factor of [latex]c[/latex]. If [latex]c >1[/latex], the graph of [latex]f(cx)[/latex] is the graph of [latex]f(x)[/latex] compressed horizontally. If [latex]0 < c < 1[/latex], the graph of [latex]f(cx)[/latex] is the graph of [latex]f(x)[/latex] stretched horizontally. For example, consider the function [latex]f(x)=\sqrt{2x}[/latex] and evaluate [latex]f[/latex] at [latex]x/2.[/latex] Since [latex]f(x/2)=\sqrt{x}[/latex], the graph of [latex]f(x)=\sqrt{2x}[/latex] is the graph of [latex]y=\sqrt{x}[/latex] compressed horizontally. The graph of [latex]y=\sqrt{x/2}[/latex] is a horizontal stretch of the graph of [latex]y=\sqrt{x}[/latex] ( (Figure) ).

We have explored what happens to the graph of a function [latex]f[/latex] when we multiply [latex]f[/latex] by a constant [latex]c >0[/latex] to get a new function [latex]cf(x)[/latex]. We have also discussed what happens to the graph of a function [latex]f[/latex] when we multiply the independent variable [latex]x[/latex] by [latex]c >0[/latex] to get a new function [latex]f(cx)[/latex]. However, we have not addressed what happens to the graph of the function if the constant [latex]c[/latex] is negative. If we have a constant [latex]c < 0[/latex], we can write [latex]c[/latex] as a positive number multiplied by -1; but, what kind of transformation do we get when we multiply the function or its argument by -1? When we multiply all the outputs by -1, we get a reflection about the [latex]x[/latex]-axis. When we multiply all inputs by -1, we get a reflection about the [latex]y[/latex]-axis. For example, the graph of [latex]f(x)=-(x^3+1)[/latex] is the graph of [latex]y=(x^3+1)[/latex] reflected about the [latex]x[/latex]-axis. The graph of [latex]f(x)=(-x)^3+1[/latex] is the graph of [latex]y=x^3+1[/latex] reflected about the [latex]y[/latex]-axis ( (Figure) ).

If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function [latex]f(x)[/latex], the graph of the related function [latex]y=cf(a(x+b))+d[/latex] can be obtained from the graph of [latex]y=f(x)[/latex] by performing the transformations in the following order.

1. Horizontal shift of the graph of [latex]y=f(x)[/latex]. If [latex]b >0[/latex], shift left. If [latex]b < 0[/latex], shift right.
2. Horizontal scaling of the graph of [latex]y=f(x+b)[/latex] by a factor of [latex]|a|[/latex]. If [latex]a < 0[/latex], reflect the graph about the [latex]y[/latex]-axis.
3. Vertical scaling of the graph of [latex]y=f(a(x+b))[/latex] by a factor of [latex]|c|[/latex]. If [latex]c < 0[/latex], reflect the graph about the [latex]x[/latex]-axis.
4. Vertical shift of the graph of [latex]y=cf(a(x+b))[/latex]. If [latex]d >0[/latex], shift up. If [latex]d < 0[/latex], shift down.

We can summarize the different transformations and their related effects on the graph of a function in the following table.

0)” and has the values “f(x) +c; f(x) -c; f(x + c); f(x – c); cf(x); f(cx); -f(x); f(-x)”. The second column is labeled “Effect on the graph of f” and the values are “Vertical shift up c units; Vertical shift down c units; Shift left by c units; Shift right by c units; ‘Vertical stretch if c >1, Vertical compression is 0 < c < 1′; ‘Horizontal stretch if 0 < c 1′; reflection about the x-axis; reflection about the y-axis”.