Sketch a graph of f(x) = -3 - e^(x+5) Find the y intercept and the horizontal asymptote of the graph. State whether the function is increasing or decreasing.

Answer 1

Remembering that the horizontal asymptote is looking at the limit of the function as the independent variable approaches +/- infinity.

The y-intercept is, by definition, when x = 0 evaluated by the function.

f(x) = -3 - e^(x+5)

f(0) = y-intercept --> -3 - e^(0+5) = -3 - e^5 ~ -151.41 --> (0, -151.41)

Now lets look at the horizontal asymptotes...

lim_{x --> -∞} f(x) = -3 - e^(-∞ + 5) = -3 - 0 = -3 [ horizontal asymptote heading toward negative infinity ]

lim_{x -->∞} f(x) = -3 - e^(∞ + 5) = -∞ [ no horizontal asymptote heading toward positive infinity ]

The function is decreasing from (-∞, ∞) due to the exponential function.