Problem 32 - College Algebra with Corequisite Support

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 \(f(x) 4\) units upward

Step-by-Step Solution

Answer

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

Step 1: Understand the transformation

We want to shift the graph of the function upward by 4 units. This means we will be adding 4 to the output of the function for each input value.

Step 2: Write the transformed function

Start with the original function, which is \( f(x) = 4^x \). To shift this function upward by 4 units, we add 4 to it. Thus, the new function is \( g(x) = 4^x + 4 \).

Key Concepts

Understanding Exponential Functions

Exponential functions are a foundational concept in mathematics. They take the form of \( f(x) = a^x \), where \( a \) is a positive real number, and \( a eq 1 \). In this kind of function, the variable \( x \) is an exponent. It's essential because exponential functions model many real-world phenomena, such as population growth or radioactive decay. For our exercise, the base is 4, which means the function grows exponentially as \( x \) increases. Compared to a linear function, exponential functions grow much faster. Understanding the base is crucial:

• A larger base leads to a faster growth rate of the function.
• The base of 4 means that every one-unit increase in \( x \) multiplies the output by 4.

The graph of \( f(x) = 4^x \) is a smooth curve that starts from a point very close to zero when \( x \) is negative, shoots up rapidly as \( x \) turns positive, reflecting the increasing speed of growth.

Graph Shifts and Their Impact

Graph shifts are common transformations in functions and are essential for understanding how graphs can be moved around on a coordinate plane. A graph shift happens when you move the entire graph of a function either up, down, left, or right.In our exercise:

• We are interested in an upward shift.
• This is achieved by adding a constant to the function. The constant amount represents the units the graph is moved.

In this case, shifting up by 4 units results in the new function \( g(x) = 4^x + 4 \). Here’s how it works:- Originally, every point on the graph \((x, f(x))\) gets transformed.- Each point goes higher by 4 units, i.e., \((x, f(x) + 4)\), because the value of \( f(x) \) is just increased by 4.Visualizing these shifts helps tremendously in understanding function behavior and predicting effects of additional transformations.

Function Transformations

The transformation of functions is a key concept that involves systematically changing the functions to shift, reflect, stretch, or compress their graphs. Transformations allow us to see the effects of mathematical operations directly on the graph layout. There are four primary transformation types:

• Translation: involves shifting the graph up, down, left, or right.
• Reflection: involves flipping the graph over a line, like the x-axis or y-axis.
• Dilation (Stretch/Compression): involves making the graph narrower or wider, or making it taller or shorter.
• Rotation: though not typically used in basic transformations, involves rotating the graph about a point.

Our exercise covers a vertical translation. The clean addition of a constant, like in \( g(x) = 4^x + 4 \), indicates a simple upward shift. Recognizing and applying these transformations helps in manipulating graphs for desired outcomes. Knowing how to combine multiple transformations is also useful, but each must be managed one step at a time to maintain clarity.