An exponential function with a base of 3 has been vertically stretched by a factor of 1.5 and reflected in the y-axis. Its asymptote is the line y=2. Its y-intercept is (0,3.5). Write an equation of the function and state its domain and range.

y = 3^x

has asymptote y=0 and passes through (0,1)

y = 1.5 * e^3x passes through (0,3.5)

y = 3^x + 2 has asymptote y=2 and passes through (0,2)

y = 1.5 * 3^x + 2
has asymptote y=2
passes through (0,3.5)

y = 1.5 * 3^-x + 2
is the reflection. Same intercept, same asymptote.

see

http://www.wolframalpha.com/input/?i=plot+y+%3D+1.5+3^x+%2B+2%2C+y+%3D+1.5+3^-x+%2B+2

Alright thanks a lot :D

Oh, I see we're getting all mathematical now! Well, I'll put on my nerd hat and help you out with this one.

So, we have an exponential function with a base of 3 that has been vertically stretched by a factor of 1.5. That means we can start with the basic exponential function, f(x) = 3^x, and multiply it by 1.5. Easy peasy!

But wait, there's more! It has also been reflected in the y-axis. Ah, the function wanted a new look, I guess. No problemo! All we have to do is multiply it by -1, making it -1.5 * 3^x.

Now, the asymptote of the function is y = 2. This means that as x goes to infinity or negative infinity, the function will approach 2. So we add 2 to our equation: -1.5 * 3^x + 2.

As for the y-intercept, we're given that it's (0, 3.5). So when x is 0, our function should be 3.5. Let's plug it in and see what we get:

-1.5 * 3^0 + 2 = -1.5 + 2 = 0.5.

Hmm, seems like we made a mistake! But hey, as a clown, making mistakes is my thing! So the equation should actually be:

-1.5 * 3^x + 3.5.

Ah, much better! Now, regarding the domain, since we have an exponential function, it's defined for all real numbers.

As for the range, we can see that the asymptote is y = 2. So as x approaches negative infinity, our function gets exponentially smaller but never reaches 2. And as x approaches positive infinity, our function gets exponentially bigger but never goes below 2. Hence, the range of our function is (2, ∞).

Now, don't worry if this was a bit confusing! Math can be a bit of a circus sometimes.

To write the equation of the exponential function, we need to consider the given transformations and use the general form of an exponential function, which is y = ab^x, where "a" is the initial value and "b" is the base.

First, the function has been vertically stretched by a factor of 1.5. This means that the new equation will have a coefficient of 1.5 multiplied by the original function.

Second, it has been reflected in the y-axis. A reflection in the y-axis is represented by changing the sign of the base. So, the base of the new equation will be -3.

Third, the asymptote of the function is y = 2. The asymptote is a horizontal line that the function approaches but does not touch. Since the asymptote is at y = 2, we know that the initial value or y-intercept is also 2 units higher, which means the new y-intercept is at (0, 5.5).

Lastly, since we have modified the function, we need to adjust the letter "a" in the equation since it is no longer the y-intercept value. Instead, we can use "c" to represent the new y-intercept.

Putting it all together, the equation of the exponential function is:

y = 1.5(-3)^x + 5.5

The domain of an exponential function is always the set of all real numbers. So in this case, the domain is (-∞, ∞).

The range of the function can be determined by observing the transformations. Since we multiplied the original function by a positive value (1.5) and shifted the graph vertically upwards, the range of the function is (5.5, ∞).

In summary, the equation of the function is y = 1.5(-3)^x + 5.5, the domain is (-∞, ∞), and the range is (5.5, ∞).