Let k(x) = -3.7e^0.8x + 8.2
Find the asymptote, y-intercept, k(-2), and k(6.25)
1. Asymptote =
2. Y - intercept =
3. k(-2) = (leave answer in terms of e)
4. k(6.25) = (leave answer in terms of e)
since e^x -> 0 for large negative x, the asymptote lies at k=8.2
The k-intercept, where x=0 is very easy, since e^0 = 1.
I'll do #3; maybe you can do #4 in like wise:
k(-2) = -3.7e^(0.8*2) + 8.2 = -3.7e^1.6 + 8.2
Would the y-intercept in this case be 1?
To find the asymptote and y-intercept of the function k(x), we need to analyze the equation.
1. Asymptote:
The asymptote of a function is a line that the graph of the function approaches but does not touch. In this case, we can determine the asymptote by looking at the behavior of the exponential term as x approaches infinity or negative infinity.
By observing the term -3.7e^0.8x, we can see that as x approaches negative infinity, e^0.8x will approach 0, since the base of the exponential term is e (approximately 2.71828), and any negative exponent of e will approach zero. Therefore, -3.7e^0.8x will approach zero as well, and the asymptote will be y = 8.2.
2. Y-intercept:
The y-intercept is the point at which the graph of the function intersects the y-axis. We can find it by substituting x = 0 into the equation k(x) = -3.7e^0.8x + 8.2:
k(0) = -3.7e^0.8(0) + 8.2
k(0) = -3.7e^0 + 8.2
k(0) = -3.7(1) + 8.2
k(0) = -3.7 + 8.2
k(0) = 4.5
So the y-intercept is the point (0, 4.5).
3. k(-2):
To find k(-2), we substitute x = -2 into the equation:
k(-2) = -3.7e^0.8(-2) + 8.2
k(-2) = -3.7e^-1.6 + 8.2
Now we can use the property e^(-a) = 1/e^a to simplify the equation:
k(-2) = -3.7/e^1.6 + 8.2
k(-2) = -3.7/e^(8/5) + 8.2
Therefore, k(-2) = -3.7/e^(8/5) + 8.2.
4. k(6.25):
To find k(6.25), we substitute x = 6.25 into the equation:
k(6.25) = -3.7e^0.8(6.25) + 8.2
k(6.25) = -3.7e^5 + 8.2
Therefore, k(6.25) = -3.7e^5 + 8.2.
Note that both k(-2) and k(6.25) are given in terms of e, because the base of the exponential function e is involved in the calculations.