Exponentials
By DarthVader
Date: 2022-06-07
Topic: 119 see comments
Post views: 1103
Exponentials
Interactive Graph:
https://learn2.open.ac.uk/mod/oucontent/view.php?id=1906295&extra=thumbnail_idm206
Exponential Growth and Decay
y = abx ⇢ this is the same as: f(x) = abx
y = abx explained:
y = the value of the y axis which is: (a × b raised to the power of the value of the x-axis (x)
a = the y-intercept
b = the scale factor (multiplication factor)
x = the value of the x-axis
Exponential Growth and Decay
A variable y is said to change exponentially with respect to variable x if the relationship between x and y is given by an equation of the form:
y = abx
where a and b are positive constants, with b not equal to 1.
If b > 1, then y grows exponentially.
The higher the value of the scale factor b, the steeper the exponential curve will be on a graph.
If 0 < b < 1, then y decays exponentially.
With a scale factor between 1 and 0, the exponential curve will slope down towards the x axis.
Discrete and Continuous
Discrete:
If the change happens in steps (x takes values from a range of equally spaced number, such as the non-negative integers), then it is discrete exponential change.
Continuous:
If the change happens continuously (x takes values from an interval of real numbers, such as the non-negative real numbers), then it is continuous exponential change.
Exponential Growth
A formula for exponential growth with any initial value and any scale factor can be worked out if the initial value is a and the scale factor is b, then:
- after 1 step, the value is a × b = ab
- after 2 steps, the value is a × b × b = ab2
- after 3 steps, the value is a × b × b × b = ab3
and so on. In general, after n steps, the value y is given by:
y = abn
The word exponential arises from the fact that the number of steps, n, is the exponent in the formula y = abn.
Alternative form for exponential equations
Any exponential equation y = bx, where b is a positive constant not equal to 1, can be written in the alternative form:
y = aekx
where k is a non-zero constant. The constant k is given by: k = ln b
a is also a constant
So if you want to create an exponential model for a situation of exponential growth or decay, you can plug some known values for x and y, then solve simultaneous equations to solve for the constants k and a.
You can then create an exponential equation that shows the values for x and y at all the points on the graph.
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