Many functions containing a variable exponent, are called exponential functions. Formally, any function that can be written in the following form is an exponential function.
Here, the coefficient, is the -intercept, which is sometimes referred to as the initial value. The base, can be interpreted as the constant multiplier. To ensure that is an exponential function, there are restrictions on and
Graph the function
The points found above all lie on the function. To graph the function, we can plot them in a coordinate plane and connect them with a smooth curve.
The initial value, of an exponential function is the number without an exponent. In this case, The constant multiplier, is the number with the exponent. Here,
When the -value increases by the -value is multiplied by Since the -value when is Thus, also lies on the graph of the function. Similarly, the point lies on the graph because These points are shown on the graph.
This process can be repeated until a general form of the graph emerges.
Lastly, the graph can be drawn by connecting the points with a smooth curve.
In scientists discovered a rare population of Flemish Giant rabbits in a secluded forest. Since then, they've been monitoring the population. During the five years of the study, the number of rabbits could be modeled with the exponential function shown.
Use the graph to write the rule for the function, then interpret its initial value and constant multiplier.
To write an exponential function rule, we need the initial value of the function, and the constant multiplier, Notice that the graph starts at This means that is the initial value.
Since we can write the following incomplete function rule. To determine we can use another point on the graph.
If the dependent variable of an exponential function written in the form is exchanged for a constant, say the result is a one-variable equation:This type of equation is called an exponential equation, and can be solved graphically. This is done by first graphing the function then finding the -coordinate of the point(s) on the graph with the -coordinate The -coordinate(s) is the solution to the equation.
Use the graph to solve the equation
The graph shows all - points that satisfy the function rule Let's compare the function rule and the equation. The only difference between these two equalities is that the independent variable, is replaced by a in the equation. Thus, we solve the equation by finding the -coordinate of any point on the graph that has the -coordinate
We can identify one such point in the graph. Let's now find the -coordinate of this point graphically.
This -coordinate is not easily read from the graph, so we'll have to make an approximation. It's just a bit bigger than so we'll use This means that an approximate solution to the equation is We can verify this by substituting it into equation to see if a true statement is made.
The right-hand side and the left-hand side are approximately equal, so we have indeed found an approximate solution to the equation: