Which Steps Will Translate $f(x)=3^x$ To $g(x)=3^{x+1}+4$?A. Shift \$f(x)=3^x$[/tex\] One Unit Up And Four Units To The Right.B. Shift $f(x)=3^x$ One Unit Up And Four Units To The Left.C. Shift
Introduction
In mathematics, functions are used to describe relationships between variables. When we transform a function, we are essentially changing its form while maintaining its underlying structure. In this article, we will explore the steps required to transform the function $f(x)=3^x$ into $g(x)=3^{x+1}+4$. We will examine the different types of transformations, including shifts, and provide a step-by-step guide on how to perform them.
Understanding the Original Function
The original function is $f(x)=3^x$. This is an exponential function, where the base is 3 and the exponent is x. The graph of this function is a curve that increases rapidly as x increases.
Understanding the Target Function
The target function is $g(x)=3^{x+1}+4$. This function is also an exponential function, but with a base of 3 and an exponent of x+1. The graph of this function is a curve that increases rapidly as x increases, but it is shifted to the left compared to the original function.
Step 1: Identify the Type of Transformation
To transform the original function into the target function, we need to identify the type of transformation required. In this case, we need to shift the original function to the left by one unit and then add 4 to the result.
Step 2: Shift the Original Function to the Left
To shift the original function to the left by one unit, we need to replace x with x+1 in the original function. This gives us:
Step 3: Add 4 to the Result
To add 4 to the result, we need to add 4 to the expression:
Step 4: Simplify the Expression
The expression $3^{x+1}+4$ is already simplified, so we can stop here.
Conclusion
In conclusion, to transform the function $f(x)=3^x$ into $g(x)=3^{x+1}+4$, we need to shift the original function to the left by one unit and then add 4 to the result. This can be achieved by replacing x with x+1 in the original function and then adding 4 to the result.
Answer
The correct answer is:
A. Shift $f(x)=3^x$ one unit up and four units to the left.
Note
The other options are incorrect because:
- Option B is incorrect because we need to shift the function to the left, not to the right.
- Option C is incorrect because we need to add 4 to the result, not shift the function up.
Final Thoughts
Transforming functions is an important concept in mathematics, and it has many real-world applications. By understanding how to transform functions, we can solve a wide range of problems in fields such as physics, engineering, and economics. In this article, we have provided a step-by-step guide on how to transform the function $f(x)=3^x$ into $g(x)=3^{x+1}+4$. We hope that this guide has been helpful in understanding the concept of function transformation.