Problem 58 - Algebra 2

Question

Describe how the graph of each function relates to the graph of \(y=4(0.8)^{x}\) $$ y=4(0.8)^{x}+3 $$

Step-by-Step Solution

Answer

The graph of the function \(y=4(0.8)^{x}+3\) is identical to the base function \(y=4(0.8)^{x}\), but it has been shifted 3 units upward.

Step 1: Identify the Base Function and New Function

The base function is \(y=4(0.8)^{x}\), while the new function is \(y=4(0.8)^{x}+3\)

Step 2: Analyze the Change

The change in the new function from the base function is an addition of '+3'.

Step 3: Interpret the Change

The '+3' indicates a vertical shift in the graph. Specifically, it moves the graph upward by 3 units.

Key Concepts

Function transformations

Function transformations are changes applied to a base function to modify its graph. These changes help in understanding how different parts of a function affect its graph's position and shape. Transformations include translations (shifts), reflections, stretches, and compressions. For exponential functions, such as the function in the original exercise, transformations primarily modify how the curve increases or decreases.

To illustrate, consider the base function \(y=4(0.8)^x\). This graphically represents an exponential decay because the base of the exponent, 0.8, is less than 1. Transforming this function does not change the base behavior of decay, but alters its position on the graph.

In our example, the transformation applied is a vertical shift, which we will explore further. It's important to remember that even small transformations can significantly change the look and interpretation of a graph.

Vertical shifts

Vertical shifts occur when a constant is added or subtracted from a function's equation, moving the entire graph up or down. This transformation does not affect the shape or orientation of the graph.

In our specific case, for the original function \(y=4(0.8)^x\), the transformation \(+3\) leads to the new function \(y=4(0.8)^x+3\). This addition is known as a vertical shift upwards by 3 units. It affects the output of the function:

• Original y-values are increased by 3.
• Each point on the graph moves vertically without any horizontal or other distortion.

Such transformations help analyze how changes to function constants affect output values and graph positioning, making it easier to predict how the function will behave after being transformed.

Graph analysis

Graph analysis involves examining the elements and transformations in a function to understand the overall behavior of the function's graph. This is essential, especially for functions that are used to model real-world scenarios.

In analyzing the graph of \(y=4(0.8)^x+3\), notice the core components:

• **Base:** The decline shown by \(0.8^x\) where each increase in \(x\) results in a smaller output.
• **Multiplier (4):** Scales the exponential graph, making the changes in y more pronounced compared to a base multiplier of 1.
• **Vertical Shift (+3):** Directly moves the entire graph up by three units, adjusting the baseline value of the function.

By carefully analyzing these parts, one can understand how exponential decay functions behave, and how any alterations in the constants produce vertical shifts, turning these functions into a powerful tool for modeling and predicting real-world behaviors. This kind of clear analysis is essential for translating mathematical models into practical applications.