Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function.\(h(x)=4^{x-3}\)

Step-by-Step Solution

Answer

The function \(h(x)=4^{x-3}\) is a horizontal shift of the function \(4^x\), moved 3 units to the right. Hence, the y-intercept that was originally at (0,1) for the function \(4^x\) will be at (3,1) for the function \(h(x)\).

Step 1: Understand the Original Function and the Shift

The original function in this scenario is \(f(x)=4^x\). The function \(h(x)=4^{x-3}\) is obtained by applying a horizontal shift of 3 units to the right to the graph of \(f(x)\). This is because a shift is represented by the form \(4^{x-a}\), where \(a\) is the shifting amount.

Step 2: Graph the Original Function

Start by graphing the original function \(f(x)=4^x\). Remember that exponential functions are increasing functions, which means they grow as x increases. The graph of \(f(x)=4^x\) has a y-intercept at (0,1). The graph becomes increasingly steep as \(x\) increases and increasingly flat as \(x\) decreases.

Step 3: Apply the Shift to the Original Function

Now apply a horizontal shift to the graph of \(f(x)\), 3 units to the right, to graph \(h(x)\). This means each point on the function \(f(x)\) should be shifted 3 units to the right on the x-axis. This results in the new y-intercept of the function \(h(x)\) being at (3,1). The function continues to become increasingly steep as \(x\) increases and increasingly flat as \(x\) decreases.

Key Concepts

Function Graphing

Graphing functions is an essential skill in mathematics. It helps us visualize how a particular function behaves. Function graphing involves plotting points, understanding their relations, and the overall shape of the graph. For the function \( f(x) = 4^x \), we start with an exponential graph. Exponential functions have unique characteristics:

• They grow rapidly as \( x \) increases.
• The base, in this case 4, determines how quickly the function grows.
• The graph passes through the point \((0,1)\), because any number to the power of zero is one.
• The curve approaches, but never actually touches, the x-axis as it moves towards negative infinity. This is called the asymptote.

When plotting the exponential function \( f(x) = 4^x \), these features help predict its shape. As \( x \) increases, the values of \( y \) increase rapidly, creating a steep upward curve. Understanding these characteristics is crucial for accurate function graphing. Remember, the ability to visualize these shifts prepares us for more complex transformations.

Horizontal Shifts

Horizontal shifts occur when every point of a function's graph is moved left or right by a certain number of units. For \( h(x) = 4^{x-3} \), there is a horizontal shift to the right by 3 units. Here's how it works:

• A horizontal shift is determined by the subtraction inside the function's exponent, such as \( x-3 \).
• Every point on the function \( f(x) = 4^x \) will move 3 units to the right on the x-axis.
• The y-intercept of \( f(x) = 4^x \) at \((0,1)\) moves to \((3,1)\).

This transformation does not affect the overall shape or direction of the graph. It simply relocates it along the x-axis. These shifts are useful for modifying functions without altering their fundamental properties. It's particularly handy in determining how functions can be adjusted to fit different scenarios.

Exponential Growth

Exponential growth is a pattern where quantities increase over time. This happens at a rate proportional to their current value, leading to a rapid increase. With \( f(x) = 4^x \), each step in \( x \) multiplies the previous value by 4.

• The base (4 in this case) dictates the growth rate. Larger bases lead to faster growth.
• Exponential growth functions always have a positive slope, meaning they will continue to increase as \( x \) increases.
• Such functions are widely applicable in real-life instances, such as population growth, compound interest, and radioactive decay.

The nature of exponential functions like \( h(x) = 4^{x-3} \) ensures that even with horizontal shifts, their growth rate remains constant. Understanding these principles enhances comprehension of both mathematical theories and their practical applications. Exponential growth is a powerful model due to its simplicity and ability to capture complex real-world phenomena.