To sketch the graph of the function $f(x) = 5 \cdot 2^{x - 2} + 3$, we can break down the transformations from the basic exponential function $g(x) = 2^x$.

Steps to Transform the Graph

1. Start with the Basic Exponential Function:

   • The graph of $g(x) = 2^x$ is an increasing curve that passes through the point $(0, 1)$ and approaches the x-axis asymptotically.

2. Horizontal Shift:

   • The term $2^{x - 2}$ indicates a horizontal shift to the right by 2 units. This shifts all points on the graph of $g(x)$ to the right.

3. Vertical Stretch:

   • Multiplying by 5, as in $5 \cdot 2^{x - 2}$, stretches the graph vertically by a factor of 5. For example, the point $(2, 1)$ from $g(x)$ becomes $(2, 5)$ after applying the vertical stretch.

4. Vertical Shift:

   • Adding 3 translates the graph vertically upwards by 3 units. So, the point $(2, 5)$ becomes $(2, 8)$. The horizontal asymptote shifts from $y = 0$ to $y = 3$.

Summary of Transformations

• Horizontal Shift: Right 2 units.
• Vertical Stretch: By a factor of 5.
• Vertical Shift: Upward by 3 units.

Key Points to Plot

• The original point $(0, 1)$ transforms to $(2, 5 \cdot 1 + 3) = (2, 8)$.
• The point $(1, 2)$ transforms to $(3, 5 \cdot 2 + 3) = (3, 13)$.
• The point $(2, 4)$ transforms to $(4, 5 \cdot 4 + 3) = (4, 23)$.

Asymptote

• The horizontal asymptote is at $y = 3$.

Sketching the Graph

• Start by drawing the horizontal asymptote at $y = 3$.
• Plot the key points mentioned above.
• Draw a smooth curve that approaches the asymptote on the left and rises steeply as $x$ increases.

Final Graph Description

The graph of $f(x) = 5 \cdot 2^{x - 2} + 3$ is an increasing exponential function that shifts to the right, stretches vertically, and is lifted upwards.

If you have any questions or need further details, feel free to ask!

Further Exploration Questions

1. How do other exponential functions behave under similar transformations?
2. What are the characteristics of exponential growth and decay?
3. Can you explain the effect of different coefficients on the exponential function?
4. How does the horizontal shift affect the domain and range of the function?
5. What are the applications of exponential functions in real-world scenarios?

Tip

When sketching transformations, always start with identifying key points and asymptotes before drawing the graph.