for the curve y=2(3^x)+1, determine:

the horizontal asymptote
the y-intercept
range
end behaviour
also, how would i sketch this?

y=3^x has a horizontal asymptote at y=0

so does y=2(3^x)

so, y=2(3^x)+1 has a h.a. at y=1.

what is y when x=0? y=2*1+1 = 3

range is all reals > 1

all exponentials look basically the same. Just graph y=a^x passes through (0,1) and (a,1)

Stretch it by 2 and shift it up 1.

To determine the horizontal asymptote, y-intercept, range, and end behavior of the curve y = 2(3^x) + 1, let's break down each component:

1. Horizontal Asymptote:
The horizontal asymptote is the value that the function approaches as x approaches positive or negative infinity. In this case, since the base of the exponential term is 3, the horizontal asymptote of this curve is y = 1. This means that as x gets very large (positive or negative), the curve approaches a y-value of 1.

2. Y-Intercept:
The y-intercept is the point where the curve crosses the y-axis, which occurs when x = 0. To find the y-intercept, substitute x = 0 into the equation:
y = 2(3^0) + 1
y = 2(1) + 1
y = 2 + 1
y = 3
Therefore, the y-intercept is (0, 3).

3. Range:
The range of a function represents all possible y-values that the function can take. In this case, since 3^x is always positive (as x varies), the entire term 2(3^x) is always positive. Adding 1 to the positive values yields positive y-values. Therefore, the range of this function is all real numbers greater than 1.

4. End Behavior:
The end behavior of a function describes how the function behaves as x approaches positive or negative infinity. In this case, as x approaches positive infinity, the exponential term 3^x grows without bound, causing the function to increase without bound. As x approaches negative infinity, the exponential term 3^x approaches 0, which means the function approaches 1. Therefore, the end behavior is as follows:
- As x approaches positive infinity, y approaches positive infinity.
- As x approaches negative infinity, y approaches 1.

To sketch this curve, follow these steps:
1. Plot the y-intercept at (0, 3).
2. Choose several x-values to calculate corresponding y-values. For example, select x = -1, 0, 1, and 2.
3. Plot the obtained points on the graph.
4. Sketch the curve by connecting the plotted points smoothly, keeping in mind the end behavior and any other characteristics mentioned above.

Remember, if you have access to graphing software or online graphing tools, plotting the function will give you a visual representation for better accuracy.