Problem 102 - 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 \(f(x) 2\) units left

Step-by-Step Solution

Answer

The function after shifting 2 units left is \( f(x) = 4^{x+2} \).

Step 1: Understand the function transformation

The function given is \( f(x) = 4^x \). To shift a function \( f(x) \) horizontally, the transformation is applied inside the function's argument (\( x \)). In this case, we need to shift the function 2 units to the left.

Step 2: Apply the horizontal shift transformation

A horizontal shift to the left by 2 units is achieved by replacing \( x \) with \( x + 2 \). This means the function after the transformation will be \( f(x + 2) \).

Step 3: Write the resulting function after transformation

Substitute \( x + 2 \) into the original function \( f(x) = 4^x \), giving us the new function \( f(x) = 4^{x+2} \). This represents the graph of \( f(x) \) shifted 2 units to the left.

Key Concepts

Exponential Function

Understanding the exponential function is key when dealing with function transformations. An exponential function takes the form \( f(x) = a^x \), where \( a \) is a constant greater than zero. These functions display a constant percent rate of growth or decay. The base, \( 4 \), in this case, tells us by what factor the function changes when \( x \) increases by 1.
Exponential functions differ from other functions such as linear or quadratic functions in terms of growth rate.

• Rapid Growth: Exponential functions grow much faster as \( x \) increases.
• Constant Ratio: Any subsequent value of the function is a multiple of the previous value by the factor \( a \).

In our specific case, \( f(x) = 4^x \), the function has an exponential growth because the base, \( 4 \), is greater than 1, showing that the graph will rise as \( x \) gets larger.

Horizontal Shift

A horizontal shift in functions allows us to move the graph along the x-axis without altering its shape. To shift a function horizontally, adjustments are made to the inside of the function's argument, such as \( x \) in \( f(x) \).
For a leftward shift, we increase \( x \) by the shift magnitude. Conversely, for a rightward shift, \( x \) is decreased by the same magnitude. Here, a 2-unit left shift is performed on the function \( f(x) = 4^x \).

• Left Shift: Replace \( x \) with \( x + 2 \)
• New Function: \( f(x) = 4^{x+2} \)

This transformation results in the graph of \( f(x) \) moving two units to the left, keeping all other graph properties intact.

Graphing Functions

Graphing functions provides a visual understanding of how functions behave and transform under various operations. Each function has a unique graph that aids in predicting output values and transformations.
For an exponential function like \( f(x) = 4^x \), the graph is an increasing curve sweeping upwards. The horizontal shift we applied moves this curve along the x-axis, changing the function's intercept but not its fundamental growth factor or shape.

• X-axis Intercept: The shift adjusts where the function crosses the axes
• Invariant Shape: Despite shifts, the overall exponential curve remains the same

Through graphing, transformations such as leftward or rightward shifts become more transparent, helping us visualize how the function changes.

Function Notation

Function notation is a standardized way to express and evaluate functions. It simplifies describing transformations, especially when dealing with mathematical operations like shifts.
In the notation \( f(x) \), "\( f \)" represents the function name, while "\( x \)" displays the function input. This notation makes it easy to apply transformations using symbols like \( x + 2 \), denoting input changes leading to shifts.

• Modification: Adjust inputs (like \( x \)) to apply transformations
• Clarity: Maintains clear representation even in complex operations

In our discussion, replacing \( x \) with \( x + 2 \) inside \( f(x) \) efficiently shows a leftward shift operation, making the manipulation of functions clear and consistent.