An exponential function is a function with the general form f(x) = y = ab^{x} and the following conditions:
Notice the use of the independent variable (x) as an exponent. This is important in order to have an exponential function. The input value must be an exponent!
Why a cannot be equal to 0?
If a = 0, then y = 0 × b^{x} = 0 since zero times anything is zero.
Therefore, a cannot be zero since it will make the function equal to zero.
Why b must be bigger than zero?
In other words, b cannot be zero and b cannot be a negative number.
b cannot be zero since y will be equal to a × 0^{x} = a × 0 = 0
b cannot be negative either. This can create some problems. For example, suppose b = -1, we get y = a (-1)^{x}
When x = 0.5, y = a(-1)^{0.5} = a √(-1) and √(-1) is a complex number.
Why b cannot equal to 1?
If b = 1, then y = a × 1^{x} = a × 1 = a since 1 to any power is equal to 1 and a times 1 is a.
Notice that x disappears as an exponent when b = 1. Therefore, b cannot be equal to 1.
When b > 1, you can model growth and b is the growth factor.
When 0 < b < 1, you can model decay and b is the decay factor.
1. y = 0.5 × 2^{x}
2. y = -3 × 0.4^{x}
3. y = e^{x}
4. y = 10^{x}
5. y = 8(1/5)^{x}
Can you tell what b equals to for the exponential functions above? Which functions model growth? Which functions model decay?
For 0.5 × 2^{x}, b = 2
For y = -3 × 0.4^{x}, b = 0.4
For e^{x}, b = e and e = 2.71828
For 10^{x}, b = 10
y = 8(1/5)^{x}, b = 1/5
Therefore, if you graph 0.5 × 2^{x}, e^{x}, and 10^{x}, the resulting graphs will show exponential growth since b is bigger than 1.
However, if you graph -3 × 0.4^{x} and 8(1/5)^{x} the resulting graph will show exponential decay since b is between 0 and 1.
As you can see from the figure above, the general shape or graph of an exponential function can either show a growth or a decay.
If there is exponential growth, as shown in the figure on the left, the graph will curve upward. A real life example of exponential growth is population growth.
If there is exponential decay, as shown in the figure on the right, the graph will curve downward. A real life example of exponential decay is radioactive decay.
The graph crosses the y-axis, but not the x-axis.
If y = ab^{x}, a > 0 b > 0, the exponential graph has the following properties:
The graph is increasing
Domain and range
The domain is all real numbers or (-∞, ∞)
The range is all positive real numbers (0, ∞)
End behavior
As x approaches negative infinity, the graph will get closer and closer to the x-axis. Therefore, the graph has a horizontal asymptote that is equal to y = 0.
As x approaches positive infinity, the graph will increase without bound. This means that y will also approach infinity.
Intercepts
The x-intercept does not exist since the graph never crosses x-axis.
The y-intercept is (0,a) since y = ab^{0} = a(1) = a. For example, if the exponential function is y = 3^{x}, the y-intercept is (0,1).
If y = ab^{x}, a > 0 0 < b < 1, the exponential graph has the following properties:
The graph is decreasing
Domain and range
The domain is all real numbers or (-∞, ∞)
The range is all positive real numbers (0, ∞)
End behavior
As x approaches negative infinity, the graph will increase without bound. This means that y will also approach infinity.
As x approaches positive infinity, the graph will get closer and closer to the x-axis. Therefore, the graph has a horizontal asymptote that is equal to y = 0.
Intercepts
The x-intercept does not exist since the graph never crosses x-axis.
The y-intercept is (0,a) since y = ab^{0} = a(1) = a. For example, if the exponential function is y = 3^{x}, the y-intercept is (0,1).
Let c be a positive real number. There are basically 5 types of transformations that can happen with exponential functions using f(x) = b^{x} as the parent function.
The vertical translation or vertical shift occurs when the constant c is added to or subtracted from the parent function.
If c is added to the parent function, then b^{x} + c shifts the graph of f(x) upward c units.
If c is subtracted from the parent function, then b^{x} - c shifts the graph of f(x) downward c units.
The horizontal translation or horizontal shift occurs when the constant c is added to or subtracted from the input value of the parent function.
If c is added to the input value of the parent function, then b^{x+c} shifts the graph of f(x) to the left c units.
If c is subtracted from the input value of the parent function, then b^{x-c} shifts the graph of f(x) to the right c units.
Reflection occurs either with respect to the x-axis or the y-axis.
If you take the opposite of the parent function, then -b^{x} reflects the graph of f(x) about the x-axis.
If you take the opposite of the input value of the parent function, then b^{-x} reflects the graph of f(x) about the y-axis.
The vertical stretch occurs when the constant c is multiplied by the parent function.
If c is bigger than 1 and c is multiplied by the parent function, then cb^{x} vertically stretches the graph of f(x).
If 0 < c < 1 and c is multiplied by the parent function, then cb^{x} vertically shrinks or compresses the graph of f(x).
The horizontal stretch occurs when the constant c is multiplied by the input value of the parent function.
If c is bigger than 1 and c is multiplied by the input value of the parent function, then b^{cx} horizontally shrinks the graph of f(x).
If 0 < c < 1 and c is multiplied by input value of the parent function, then b^{cx} horizontally stretches the graph of f(x).
Sep 17, 23 09:46 AM
Jun 09, 23 12:04 PM