Problem 100 - 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 \(f(x)\) 4 units upward

Step-by-Step Solution

Answer

The function after shifting upward is \(g(x) = 4^x + 4\).

Step 1: Understand the Parent Function

The parent function is given as \(f(x) = 4^x\). This is an exponential function with a base of 4, which means it grows rapidly as \(x\) increases.

Step 2: Analyze the Transformation

The transformation described involves shifting the function 4 units upward. In terms of function transformation, this means adding 4 to the function's output, \(f(x)\).

Step 3: Apply the Transformation

To Shifting \(f(x)\) 4 units upward, we modify the original function by adding 4 to it. The transformed function becomes \(g(x) = 4^x + 4\).

Step 4: Verify the Transformation

Check a few values to ensure the transformation is correct. For the original \(f(0) = 4^0 = 1\) and the transformed \(g(0) = 4^0 + 4 = 5\). The output has increased by 4, confirming the correct transformation was made.

Key Concepts

Function Transformation

In the world of mathematics, especially when dealing with functions, transformations are a common and powerful tool. Transformations involve modifying a function to produce a new function. This can help us understand behaviors of graphs and functions. The basic forms of function transformations include:

• Vertical Translations: Moving a graph up or down.
• Horizontal Translations: Shifting a graph left or right.
• Reflections: Flipping a graph over a line, such as the x-axis or y-axis.
• Stretches and Compressions: Altering the size of a graph.

Function transformations can be applied to any type of function, including linear, quadratic, and exponential functions, like the one in this exercise. The function given here is an exponential function, which we often transform to analyze shifts in growth or decay when plotted on a graph. Understanding these transformations helps us predict how a graph will shift or change its shape.

Graph Shifting

Graph shifting is a specific type of function transformation that moves the graph either up, down, left, or right, without altering its shape. This exercise demonstrates an upward shift.When you shift a graph vertically, you adjust the entire graph's position in relation to the y-axis. If you want to shift it upward, you add a constant to the function. For example, shifting up by 4 means modifying the original function by adding 4.In mathematical terms, for a given function, say, \(f(x)\), the transformed function that shifts upward becomes \(g(x) = f(x) + k\), where \(k\) is the number of units shifted. In this case, since the original function is \(f(x) = 4^x\) and we're shifting it 4 units upward, the new function is \(g(x) = 4^x + 4\).This adjustment causes each point on the graph to move up by 4 units, effectively raising the entire function without changing its growth rate or shape.

Parent Function Analysis

The concept of a parent function is central to understanding more complex function transformations. A parent function is the simplest function of a family of functions. In this case, the parent function is \(f(x) = 4^x\).Analyzing a parent function involves:

• Identifying its basic shape and characteristics.
• Understanding the rate at which it increases or decreases.
• Examining key points, such as intercepts and asymptotes.

For \(f(x) = 4^x\), it's an exponential growth function:

• Its graph passes through the point (0,1) because \(4^0 = 1\).
• It gets rapidly steeper as \(x\) increases, indicating exponential growth.
• There's a horizontal asymptote at \(y = 0\).

Understanding this parent function lays the foundation for analyzing how transformations, like shifts or stretches, will alter the graph's appearance and behavior. Parent function analysis is invaluable for predicting and validating the effects of these transformations.