5. sketch the graph of $f(x) = e^{x-3} + 6$.

Explanation:

Step1: Identify the parent function and transformations

The parent function is $y = e^x$. The transformations are a horizontal shift right by 3 units and a vertical shift up by 6 units.

Step2: Determine the horizontal asymptote

As $x \to -\infty$, $e^{x-3} \to 0$, so the function approaches the constant value. $$y = 6$$

Step3: Find the y-intercept

Set $x = 0$ to find the point where the graph crosses the y-axis. $$f(0) = e^{0-3} + 6 = e^{-3} + 6 \approx 6.05$$

Step4: Find a reference point

Set $x = 3$ to simplify the exponent and find a clear coordinate. $$f(3) = e^{3-3} + 6 = e^0 + 6 = 1 + 6 = 7$$

Step5: Describe the graph shape

The graph is an increasing exponential curve starting near $y=6$ on the left and rising sharply through $(3, 7)$.

Answer:

The graph of $f(x) = e^{x-3} + 6$ is an exponential growth curve with a horizontal asymptote at $y = 6$. It passes through the points $(0, e^{-3} + 6)$ and $(3, 7)$. As $x$ increases, the graph rises infinitely; as $x$ decreases, it approaches the line $y = 6$ from above.