Question Find the $$ y $$-intercept and equation of the horizontal asymptote of the function $$ f(x)=-3 e^{x+2}+4 $$. Round to the the nearest two decimal places as needed.

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UpStudy AI · Accepted Answer

**Step 1. Find the $$ y $$-intercept**

The $$ y $$-intercept occurs when $$ x = 0 $$. Substitute $$ x = 0 $$ into the function:
$$
f(0) = -3e^{0+2} + 4 = -3e^2 + 4.
$$
A numerical approximation using $$ e^2 \approx 7.39 $$ gives:
$$
f(0) \approx -3(7.39) + 4 = -22.17 + 4 = -18.17.
$$
Thus, the $$ y $$-intercept is approximately $$(-18.17)$$ (i.e., the point $$(0, -18.17)$$).

**Step 2. Determine the equation of the horizontal asymptote**

For the function
$$
f(x) = -3e^{x+2} + 4,
$$
we consider the limits as $$ x $$ approaches both $$ \infty $$ and $$ -\infty $$.

- As $$ x \to \infty $$, $$ e^{x+2} $$ grows without bound, so
  $$
  -3e^{x+2} + 4 \to -\infty.
  $$
  There is no horizontal asymptote in this direction.

- As $$ x \to -\infty $$, $$ e^{x+2} \to 0 $$ because the exponential decays to zero. Therefore,
  $$
  f(x) \to -3(0) + 4 = 4.
  $$
  This means the horizontal asymptote is:
  $$
  y = 4.
  $$

**Final Answers:**

- The $$ y $$-intercept is $$ \boxed{(-18.17)} $$ (as the point $$(0, -18.17)$$).
- The horizontal asymptote is $$ \boxed{y = 4} $$.
The $$ y $$-intercept is $$(-18.17)$$ and the horizontal asymptote is $$ y = 4 $$.

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Find the -intercept and equation of the horizontal asymptote of the function . Round to the the nearest
two decimal places as needed.

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Question Find the -intercept and equation of the horizontal asymptote of the function . Round to the the nearest two decimal places as needed.