Problem 101 - Precalculus

Question

For the following exercises, start with the graph of \(f(x)=4^{x}\) . Then write a function that results from the given transformation. Shift \(f(x) 3\) units downward

Step-by-Step Solution

Answer

The transformed function is \( g(x) = 4^x - 3 \).

Step 1: Understanding the Base Function

The base function given is \( f(x) = 4^x \). This is an exponential function with a base of 4, meaning that as \( x \) increases, the function grows exponentially.

Step 2: Understanding the Transformation

The exercise asks for a transformation that involves shifting the base function \( f(x) \) downwards by 3 units. This is a vertical shift of the entire graph of the function.

Step 3: Applying the Downward Shift

To shift the function \( f(x) = 4^x \) downward by 3 units, we subtract 3 from the entire function. Thus, the transformed function becomes \( g(x) = 4^x - 3 \).

Step 4: Conclusion

The function \( g(x) = 4^x - 3 \) is the result of shifting the original function downward by 3 units. This means for every point on the graph of \( f(x) = 4^x \), the corresponding point on \( g(x) \) is 3 units lower.

Key Concepts

Graph Transformations

Graph transformations are fundamental alterations that change the position, shape, or size of a graph. One common type is the exponential function transformation, like with our example.
The original function is an exponential function described by \( f(x) = 4^x \). Knowing the base graph shape, a basic transformation might include:

• Translation: Moving the graph up, down, left, or right.
• Reflection: Flipping the graph over an axis.
• Stretch or Compression: Altering the graph’s width or height.

Here, we focus on a vertical translation. This places the entire graph in a different vertical position, preserving its shape.

Vertical Shifts

A vertical shift involves adjusting the graph of a function upwards or downwards. It's a straightforward process that changes every y-coordinate of your function’s graph. For example,
in the transformation from \( f(x) = 4^x \) to \( g(x) = 4^x - 3 \), we see a vertical shift downward by 3 units.
To perform a vertical shift on a function:

• Add a constant to shift it upward. For instance, \( f(x) = 4^x + 3 \) moves the graph 3 units up.
• Subtract a constant to shift it downward, like in our original problem.

The values of \( y \) decrease by 3 at each point for \( g(x) = 4^x - 3 \), effectively moving the graph lower.

Function Transformations

Function transformations encompass all types of manipulations that can be performed on the graph of a function. They can make a simple function look entirely different without altering its basic form.
In exponential functions like \( f(x) = 4^x \), transformations include:

• Vertical and horizontal shifts.
• Reflections across the \( x ext{-axis} \) or \( y ext{-axis} \).
• Stretching or compressing the function.

Understanding function transformations helps you to anticipate how changing certain components in the equation will affect the graph. This mental map allows for graph manipulation, essential in calculus and beyond.