- Identify the horizontal shift by expressing the exponent \(x+1\) as \(x - (-1)\), which represents a shift of 1 unit to the left.
- Identify the vertical shift by observing the constant added to the function, which is +4, representing a shift of 4 units up.
- Conclude that the transformation is a shift 1 unit to the left and 4 units up.
- The correct option is: \(\boxed{\text{Shift } f(x)=3^x \text{ one unit to the left and four units up.}}\)
Explanation
- Analyzing Function Transformations
The problem asks for the transformation required to map the parent exponential function \(f(x) = 3^x\) to the target function \(g(x) = 3^{x+1} + 4\). In general, a transformation of a function \(f(x)\) to \(f(x-h) + k\) corresponds to a horizontal shift of \(h\) units and a vertical shift of \(k\) units. Specifically, if \(h > 0\), the graph shifts to the right, and if \(h < 0\), the graph shifts to the left. If \(k > 0\), the graph shifts up, and if \(k < 0\), it shifts down.
- Determining the Horizontal Shift
To determine the horizontal shift, we compare the exponents of the parent and target functions. In \(f(x) = 3^x\), the exponent is \(x\). In \(g(x) = 3^{x+1} + 4\), the exponent is \(x+1\). We can rewrite the exponent as \(x - (-1)\). According to the transformation rule, where \(h = -1\), the shift is 1 unit to the left.
- Determining the Vertical Shift
To determine the vertical shift, we examine the constant added to the exponential expression. The parent function \(f(x) = 3^x\) is increased by 4 to get \(g(x) = 3^{x+1} + 4\). Therefore, the constant \(k = 4\). Since \(k\) is positive, this indicates a vertical shift of 4 units upward.
- Conclusion
By combining the results, we find that the transformation involves shifting the graph of \(f(x) = 3^x\) one unit to the left and four units up. Therefore, the correct transformation is: shift \(f(x)=3^x\) one unit to the left and four units up.
Examples
Transformations are widely used in computer graphics and animation. For instance, if a character is represented by a function \(f(x)\) on a coordinate plane, moving that character across the screen involves applying transformations. By shifting the function horizontally and vertically—such as moving it one unit left and four units up—animators can change the character's position on the screen, creating movement and interaction in games or digital art.