describe how the graph of f(x)=2^x is transformed to graph g(x)=-2^x+1   +5

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 \( k \).

describe how the graph of f(x)=2^x is transformed to graph g(x)=-2^x+1 +5

Explore the transformations applied to the graph of f(x) = 2^x to obtain g(x) = -2^(x+1) + 5. Understand reflections, horizontal shifts, and vertical shifts in exponential functions. Ideal for students in Grades 9-11.

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