The graph of a function of the form [math]y=mx+b[/math] is a [i]line[/i]. [br]This is why all the functions of this type are named [i]linear functions[/i].[br][br]If we know the coordinates of two points of the function, [math]P_1=\left(x_1,y_1\right)[/math] and [math]P_2=\left(x_2,y_2\right)[/math], we can calculate the [i]slope[/i] [math]m[/math] of the line: [math]m=\frac{y_2-y_1}{x_2-x_1}[/math]. This is a constant value: however you choose two points on the line, the value of [i]m[/i] is always the same.

In the app below, move points [math]A[/math] and [math]B[/math], then enter in the input box the value of the slope [math]m[/math] of the line that you have defined.[br]Select [i]Check answer[/i] to get a feedback for your answer and view the solution of this exercise.[br]Deselect [i]Check answer[/i] to create a new line and calculate its slope.

If you have the equation of a linear function [math]f\left(x\right)=mx+b[/math] and the coordinates of two of its points, [math]P_1=\left(x_1,y_1\right)[/math] and [math]P_2=\left(x_2,y_2\right)[/math], you can calculate: [br]- the value [math]b[/math] of the [i]y[/i]-intercept[br]- the value [math]m[/math] of the slope, using the formula [math]m=\frac{y_2-y_1}{x_2-x_1}[/math].[br][br]Move points [i]A[/i] and [i]B[/i] in the app above, and align them vertically.[br]You will discover which is the algebraic issue that is generated by such a configuration.

An [i]exponential function[/i] is a function of the form[br][center][math]f\left(x\right)=b^x[/math][/center]where the term [math]b[/math] is called [i]base[/i], with [math]b>0[/math] and [math]b\ne1[/math], and [math]x[/math] is called [i]exponent[/i], and can be any real number.[br]

The base [math]b[/math] must be:[br][list][*][i]positive[/i]: to allow the evaluation in [math]\mathbb{R}[/math] of every real number. In fact, if we had for example [math]f\left(x\right)=\left(-2\right)^x[/math], then [math]f\left(\frac{1}{2}\right)=\left(-2\right)^{\frac{1}{2}}=\sqrt{\left(-2\right)}[/math], but this operation does not yield a real number.[/*][*][i]not[/i] 0 and [i]not[/i] 1: for those values of [math]b[/math], the exponential function degenerates to the graph of a horizontal line, respectively [math]f\left(x\right)=0^x=0[/math] and [math]f\left(x\right)=1^x=1[/math].[br][/*][/list]

The applet below allows you to interact with the graph of an exponential function.[br][br][list=1][*]Use the slider that defines the value of the [i]base [/i]to view the shape of the graph when [math]b>1[/math] or [math]b<1[/math].[/*][br][*]Select the [i]Show table[/i] checkbox to view a table of values for the displayed function: three of these values are already defined, that is [math]f\left(-1\right)[/math] (the inverse value of the base), [math]f\left(0\right)[/math] (the [i]y[/i]-intercept) and [math]f\left(1\right)[/math] (the value of the base). These are the three main points that you should always use to draw the graph of an exponential function. Choose the fourth [i]x[/i] value at which you want to evaluate the function by dragging the point on the [i]x[/i]-axis. (All the values in the table are approximated to 2 decimal places).[/*][br][*]Select the [i]Monotonicity [/i]checkbox to view and explore the formal definition of [i]increasing [/i]or [i]decreasing [/i]function applied to the current graph, by dragging the points on the [i]x[/i]-axis.[/*][br][*]Select the [i]Show[/i] [math]e^x[/math] checkbox to view the graph of the exponential function with base [math]e=2.71828...[/math], that is a mathematical constant: a not terminating decimal number that has a great importance in applied mathematics.[br][br][/*][/list]

Given an exponential function [math]f\left(x\right)=b^x[/math], with [math]b>0[/math] and [math]b\ne1[/math]:[br][list][*]the domain of the function is [math]\mathbb{R}=\left(-\infty,+\infty\right)[/math][/*][*]the range of the function is [math]\left(0,\infty\right)[/math][/*][*]the [i]y[/i]-intercept of the graph is 1[br][/*][*]the function has a horizontal asymptote at [math]y=0[/math][/*][*]the function is increasing if [math]b>1[/math], and decreasing if [math]b<1[/math][br][/*][/list]