Problem 34 - College Algebra

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 \(f(x) 2\) units left

Step-by-Step Solution

Answer

The transformed function is \(g(x) = 4^{(x + 2)}\).

Step 1: Understand the Base Function

The base function given is \(f(x) = 4^x\), which is an exponential function with a base of 4. We will use this function as the starting point for the transformation.

Step 2: Recall the Transformations of Functions

A function shift can be described as moving the graph of the function horizontally, vertically, or both. For a horizontal shift, when you want to shift the function to the left by \(c\) units, you replace \(x\) with \(x + c\) in the function \(f(x)\).

Step 3: Apply the Horizontal Shift

To shift \(f(x) = 4^x\) 2 units left, we replace \(x\) with \(x + 2\). This results in the new function: \(f(x + 2) = 4^{(x + 2)}\).

Step 4: Write the Transformed Function

The function that results from shifting \(f(x) = 4^x\) 2 units to the left is \(g(x) = 4^{(x + 2)}\). This transformation shifts the entire graph of the function to the left by 2 units.

Key Concepts

Exponential Function

An exponential function is a powerful mathematical tool used to model growth or decay, where a constant base is raised to a variable exponent. The general form of an exponential function is \(f(x) = a^{x}\), where \(a\) is a positive constant greater than 1. For this specific exercise, the exponential function is \(f(x) = 4^x\). This means the base is 4, and as \(x\) changes, the output value grows exponentially. Exponential functions are known for their rapid rise or decline, making them ideal for representing phenomena like population growth, radioactive decay, or interest calculations.

• The base, like 4 in \(4^x\), determines how steep or flat the curve is.
• These functions are continuous and always above the x-axis for a positive base.
• They do not have any peaks or valleys like polynomial functions.

Understanding exponential functions requires recognizing how their growth pattern accelerates or decelerates, hinging on the base being constant, but the variable \(x\) influences the result dramatically. This sensitivity to \(x\) is precisely what makes them so useful in representing exponential growth or decay.

Horizontal Shift

A horizontal shift is a type of transformation that slides the entire graph of a function left or right along the x-axis by a specified number of units. This type of transformation keeps the shape of the graph unchanged but changes its position on the coordinate plane. For the function \(f(x) = 4^x\), shifting it 2 units to the left involves replacing \(x\) with \(x + 2\). The resulting function, \(g(x) = 4^{(x+2)}\), reflects this leftward movement.

• A shift to the left means that for any input \(x\), the function evaluates the same as it would for the input \(x-2\).
• This transformation does not affect the rate of growth or decline of an exponential function.
• Horizontal shifts are foundational when exploring phase or positional changes in function analysis.

When handling horizontal shifts, remember that they affect only the input values \(x\), offering a straightforward way to analyze positional changes without altering the function’s inherent growth nature.

Function Transformation

Function transformation entails altering a base function's graph to represent different situations or sets of data. These transformations include shifts, reflections, stretches, and compressions. Understanding function transformations helps in predicting and visualizing changes to the base function's graph.Transformations can be divided into:

• Horizontal shifts, moving the graph left or right.
• Vertical shifts, moving the graph up or down.
• Reflections, flipping the graph over a given axis.
• Stretching or compressing, altering the graph's steepness.

In the context of the exponential function \(f(x) = 4^x\), the prior example applies a horizontal shift transformation, creating \(g(x) = 4^{(x+2)}\). This transformation is easily visualized as shifting each point on the graph 2 units left.Being adept with function transformations allows for better interpretation of real-world scenarios and mathematical models by simply adjusting basic functions to fit more precisely with observed data.

Base Function

The base function is the original form from which transformations are executed. It serves as the starting point before any transformation is applied. The exponential base function in our exercise is \(f(x) = 4^x\), which stands as the foundation for applying further shifts or alterations.Key characteristics of a base function:

• It provides a static reference that other forms of the function will be compared to.
• Each transformation directly modifies or utilizes components of this base function.
• Understanding the base function aids in predicting how transformations will affect the graph.

In transformations, the base function remains constant except for the specified parameters, ensuring that its vertical behavior remains intact while horizontal and vertical positioning can change. Recognizing how the base function behaves allows for intuitive predictions of how different function transformations will manifest visually and algebraically in new forms.