How to find the Key Point and Asymptote

[sollinton]

Homework Statement

I don't particularly need help with anyone problem, I just need a refresher on how to find the Key Point and Asymptote of an equation like the following:

f(x)=2^(x+2)+3

Y-Int=?
Key Point=?
Asymptote=?

Homework Equations

I'm pretty sure I'll only need the original equation f(x)=2^(x+3)+3 and the unmodified equation f(x)=2^(x) lthough I'm not sure if my unmodified version is right.

The Attempt at a Solution

From what I remember, to find the Y-Intercept I need to set X equal to zero, and so:

f(x)=2^(0+2)+3
f(x)=4+3
f(x)=7

And so the Y-Intercept would be (0,7)

I alos remember that finding Key Point has something to do with finding the Y-Intercept without any of the additional modifiers in the equation:

f(x)=2^(0)
f(x)=1

I'm not even sure I did that right, but assuming I did, I do not know what to do with the answer. I remember I somehow combine it with the Y-Intercept, but I simply cannot remember how.

For the Asymptote, I believe I simply take the number that modifies the X and set it equal to X:

x=2

Again, I'm not sure about this, so any help would be greatly appreciated

 

[konthelion]

Your function $$f(x)=2^{x+2}+3$$ has a horizontal asymptote. To find a horizontal asymptote such that $${\lim }\limits_{x \to \infty} f(x)$$ and $${\lim }\limits_{x \to - \infty} f(x)$$ Also, horizontal asymptotes are written as "y="

 

[Architeuthis Dux]

.

As a scientist, the key point and asymptote are important concepts in understanding the behavior of a function. The key point is the point on the graph of a function that has the highest or lowest y-value, also known as the maximum or minimum point. To find the key point, you can take the derivative of the function and set it equal to zero, then solve for x. This will give you the x-value of the key point. To find the y-value, plug the x-value into the original function.

In the given equation, f(x)=2^(x+2)+3, the key point can be found by taking the derivative: f'(x)=2^(x+2)ln(2). Setting this equal to zero and solving for x, we get x=-2. This means the key point is (-2, 3).

The asymptote of a function is a line that the graph of the function approaches but never touches. In this case, the asymptote can be found by looking at the exponent of the base 2 in the function. Since the exponent is x+2, the asymptote will be the line x=-2.

To find the y-intercept, you correctly set x=0 in the original function and solve for y, giving you the point (0, 7).

In summary, to find the key point, take the derivative, set it equal to zero, solve for x, and plug the x-value into the original function. To find the asymptote, look at the exponent of the base in the function and set it equal to x. And to find the y-intercept, set x=0 in the original function and solve for y.