To understand how the graph of $f(x) = 2^x$ is transformed into the graph of $g(x) = -2^{x+1} + 5$, let's break down the transformations applied step-by-step:

1. Reflection over the x-axis: The negative sign in front of $2^{x+1}$ indicates a reflection across the x-axis. So the graph of $f(x) = 2^x$ is flipped upside down.

2. Horizontal shift left: The expression $x+1$ in the exponent indicates a horizontal shift to the left by 1 unit. So we move every point on the graph left by 1 unit.

3. Vertical shift up: The "+5" outside the exponential function indicates a vertical shift upward by 5 units. So after reflecting and shifting horizontally, we move every point up by 5 units.

In summary, the transformations applied to $f(x) = 2^x$ to obtain $g(x) = -2^{x+1} + 5$ are:

• Reflect over the x-axis.
• Shift left by 1 unit.
• Shift up by 5 units.

Would you like more details or have any questions?

Here are five related questions:

1. How does a reflection over the y-axis affect the graph of $f(x) = 2^x$?
2. What would be the transformation if the function was $g(x) = 2^{x+1} - 5$?
3. How would a stretch or compression factor in front of $2^x$ affect the graph of $f(x) = 2^x$?
4. How does changing $f(x) = 2^x$ to $f(x) = 3^x$ affect the steepness of the graph?
5. What would happen to $g(x) = -2^{x+1} + 5$ if we replaced $x+1$ with $x-2$?

Tip: Horizontal shifts inside the exponent of an exponential function $2^{x+k}$ move the graph in the opposite direction of the sign on $k$.