Write two equations to represent the same exponential function with a y-intercept of 5 and an asymptote at y=3 . Investigate whether other exponential functions have the same properties. Use the transformations to explain your observations.
its wrong
To represent the same exponential function with a y-intercept of 5 and an asymptote at y=3, we can start with the general form of an exponential function, which is:
y = a * b^x
Where 'a' is the y-intercept, 'b' is the base, and 'x' is the independent variable.
Equation 1: y = 5 * (1/2)^x
In this equation, we have set the base 'b' equal to 1/2. Since any number raised to the power of 0 equals 1, when x is zero, we get:
y = 5 * (1/2)^0 = 5 * 1 = 5
This gives us a y-intercept of 5.
When x approaches infinity or negative infinity, the term (1/2)^x approaches zero. So, we have an asymptote at y=0. However, since we want an asymptote at y=3, we need to transform the graph.
Equation 2: y = 3 + 2 * (1/2)^x
In this equation, we have added a constant term '3' to shift the graph upwards. Now, when x is zero, we get:
y = 3 + 2 * (1/2)^0 = 3 + 2 * 1 = 3 + 2 = 5
We still have a y-intercept of 5.
When x approaches infinity or negative infinity, the term (1/2)^x still approaches zero. However, we have added the constant term '3', so the asymptote is shifted to y=3.
Now, let's investigate other exponential functions with the same properties:
If we change the base 'b' to any value between 0 and 1, we will get a decrease in the function as x increases. This means that the graph will approach the asymptote from below. However, we can still shift the graph vertically with a constant term to adjust the y-intercept and the asymptote.
If we change the base 'b' to any value greater than 1, we will get an increase in the function as x increases. This means that the graph will approach the asymptote from above. Again, we can shift the graph vertically with a constant term to adjust the y-intercept and the asymptote.
In summary, to represent the same exponential function with a y-intercept and asymptote, we can use the general form 'y = a * b^x' and adjust the values of 'a', 'b', and add a constant term to shift the graph vertically. The base 'b' determines the behavior of the graph (increasing or decreasing), while the constant term adjusts the y-intercept and the asymptote.
This is incorrect
You know that y=e^x goes through (0,1) and has an asymptote at y=0, so,
y = e^x + 3 has its asymptote at y=3.
Further, 5e^x goes through (0,5), so one function would be
5e^x + 3
Now, you can pick any number greater than 1, and the above statements hold true, so
5*2^x, 5*10^x all do the same.
In fact, that also means that
5*12^7x, 5*1.001^100x all do it to.
Since a*b^(kx) = a*e^((k lnb) x)