If you know two points that fall on a particular exponential curve, you can define the curve by solving the general exponential function using those points. In practice, this means substituting the points for y and x in the equation y = abx. The procedure is easier if the x-value for one of the points is 0, which means the point is on the y-axis. If neither point has a zero x-value, the process for solving for x and y is a tad more complicated.
Why Exponential Functions Are Important
Many important systems follow exponential patterns of growth and decay. For example, the number of bacteria in a colony usually increases exponentially, and ambient radiation in the atmosphere following a nuclear event usually decreases exponentially. By taking data and plotting a curve, scientists are in a better position to make predictions.
From a Pair of Points to a Graph
Any point on a two-dimensional graph can be represented by two numbers, which are usually written in the in the form (x, y), where x defines the horizontal distance from the origin and y represents the vertical distance. For example, the point (2, 3) is two units to the right of the y-axis and three units above the x-axis. On the other hand, the point (-2, -3) is two units to the left of the y-axis. and three units below the x-axis.
If you have two points, (x1, y1) and (x2, y2), you can define the exponential function that passes through these points by substituting them in the equation y = abx and solving for a and b. In general, you have to solve this pair of equations:
y1 = abx1 and y2 = abx2, .
In this form, the math looks a little complicated, but it looks less so after you have done a few examples.
One Point on the X-axis
If one of the x-values -- say x1 -- is 0, the operation becomes very simple. For example, solving the equation for the points (0, 2) and (2, 4) yields:
2 = ab0 and 4 = ab2. Since we know that b0 = 1, the first equation becomes 2 = a. Substituting a in the second equation yields 4 = 2b2, which we simplify to b2 = 2, or b = square root of 2, which equals approximately 1.41. The defining function is then y = 2 (1.41)x.
Neither Point on the X-axis
If neither x-value is zero, solving the pair of equations is slightly more cumbersome. Henochmath walks us through an easy example to clarify this procedure. In his example, he chose the pair of points (2, 3) and (4, 27). This yields the following pair of equations:
27 = ab4
3 = ab2
If you divide the first equation by the second, you get
9 = b2
so b = 3. It's possible for b to also be equal to -3, but in this case, assume it's positive.
You can substitute this value for b in either equation to get a. It's easier to use the second equation, so:
3 = a(3)2 which can be simplified to 3 = a9, a = 3/9 or 1/3.
The equation that passes through these points can be written as y = 1/3(3)x.
An Example from the Real World
Since 1910, human population growth has been exponential, and by plotting a growth curve, scientists are in a better position to predict and plan for the future. In 1910, the world population was 1.75 billion, and in 2010, it was 6.87 billion. Taking 1910 as the starting point, this gives the pair of points (0, 1.75) and (100, 6.87). Because the x-value of the first point is zero, we can easily find a.
1.75 = ab0 or a = 1.75. Plugging this value, along with those of the second point, into the general exponential equation produces 6.87 = 1.75b100, which gives the value of b as the hundredth root of 6.87/1.75 or 3.93. So the equation becomes y = 1.75 (hundredth root of 3.93)x. Although it takes more than a slide rule to do it, scientists can use this equation to project future population numbers to help politicians in the present to create appropriate policies.