The function provided is:

$f(x) = 5(2^x) + 3$

Step-by-step analysis:

1. Base Function: The base function is $2^x$, which is an exponential growth function. It passes through the point $(0, 1)$ and increases rapidly as $x$ increases.

2. Scaling by 5: The function is multiplied by 5, which vertically stretches the graph. Points that were originally at $y$ will now be at $5y$. So, for $x = 0$, instead of $2^0 = 1$, we now have $5(2^0) = 5$.

3. Vertical Shift by 3: The "+3" at the end means the graph shifts up by 3 units. So, the point $(0, 5)$ becomes $(0, 8)$.

4. Key Point: At $x = 0$, the function gives $f(0) = 5(2^0) + 3 = 5 + 3 = 8$. So, the graph should pass through $(0, 8)$.

Now let's match this information with the given graphs:

• Check for the y-intercept: The correct graph should have a y-intercept at $(0, 8)$ and show exponential growth after that.

Conclusion:

The fourth graph from the top matches the graph of the function $f(x) = 5(2^x) + 3$ because it passes through $(0, 8)$ and displays exponential growth consistent with the function.

Would you like a detailed explanation of how to graph such functions, or any specific questions about exponential functions?

Here are 5 related questions to expand your understanding:

1. How does the graph change if the coefficient 5 is changed to another value?
2. What happens if the base of the exponential function is smaller, like $f(x) = 5(1.5^x) + 3$?
3. How do vertical shifts affect other types of functions (e.g., quadratic)?
4. What does the graph of $f(x) = 5(2^{-x}) + 3$ look like?
5. How would horizontal shifts, like $f(x) = 5(2^{x-1}) + 3$, affect the graph?

Tip: When graphing exponential functions, always start by identifying the key points such as the y-intercept and how the function grows or decays!