Determining Transformed Exponential Function from Graph

TLDR; Determining the equation of an exponential function from a graph involving shifts, reflection, and identifying the values of 'A' and 'B'.

💡 Exponential Function Equation

The task is to determine the equation for the exponential function from the provided graph using the form F(x) = A * B^x.

The graph has a horizontal asymptote at y = +4, indicating a shift of the exponential function 4 units upwards to accommodate the asymptote.

The graph also displays exponential decay reflected across the x-axis, indicating that the value of 'A' will be negative.

📈 Identifying Points on the Function

Two points on the function, (0, 2) and (-1, -4), are identified to determine the values of 'A' and 'B'.

The point (0, 2) leads to the conclusion that F(0) = 2, and the point (-1, -4) leads to the conclusion that F(-1) = -4.

Using F(0) = 2, the value of 'A' is determined to be -2, and then using the second function value, the value of 'B' is determined.

🔢 Determining Values of A and B

By substituting F(0) = 2 into the function, the value of 'A' is determined to be -2.

Subsequently, using F(-1) = -4, the value of 'B' is calculated to be 1/4.

🔄 Using Function Values to Determine B

After determining the value of 'A', the function value F(-1) = -4 is used to determine the value of 'B'.

By solving the equation -2 * B^-1 + 4 = -4, the value of 'B' is found to be 1/4.

📝 Final Exponential Function

The final exponential function is determined to be F(x) = -2 * (1/4)^x + 4.

The base being between 0 and 1 indicates exponential decay, and the negative 'A' reflects the function across the x-axis.

The +4 in the function indicates a vertical shift of 4 units upwards.

🔄 Recognizing Transformations

The transformations of the function are recognized: exponential decay due to the base being between 0 and 1, reflection across the x-axis due to the negative 'A', and a vertical shift of 4 units due to the +4 in the function.

These transformations accurately describe the provided graph in red.

🙌 Conclusion

The process of determining the equation of the transformed exponential function from a graph is concluded, providing a comprehensive understanding of the topic.