Question

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=\left(\frac{4}{3}\right)^x\)

Short answer

The function \(y=\left(\frac{4}{3}\right)^x\) is an exponential growth function. The graph would illustrate a curve that trends upwards as x increases.

Step-by-step solution

Identify exponential growth or decay

To ascertain whether it's exponential growth or decay, we'll check the base of the exponent, \(\frac{4}{3}\). Given \(0 < b < 1\) for decay and \(b > 1\) for growth, \(\frac{4}{3} > 1\), so it is an exponential growth function.

Prepare function for graph

The x-values will be chosen, then the corresponding y-values will be calculated using the function \(y=\left(\frac{4}{3}\right)^x\). A set of points \((x, y)\) will be generated from these values, which will be used to plot the function.

Graph the function

Plot the points generated in step 2 on a graph, then draw a smooth curve through the points to create the graph of the function. It'll be an upward trending curve given it's a growth function.

Key concepts

Understanding Exponential Growth

The concept of exponential growth is a fundamental part of mathematics, especially when analyzing how quantities increase over time. In an exponential growth function, the rate of growth is continuously proportional to the current amount. This means that as the quantity becomes larger, it grows at a faster rate.

In the function given, \( y = \left( \frac{4}{3} \right)^x \), we identify that it's an exponential growth due to the base of the exponent being greater than 1. Specifically, a base \( b > 1 \) signals exponential growth. This results in the values of \( y \) increasing rapidly as \( x \) becomes larger.

Real-world examples of exponential growth include populations, investments earning interest, and certain science phenomena like bacterial growth. Understanding this can help you identify and predict behaviors in various systems.

Graphing Exponential Functions

Graphing exponential functions like \( y = \left( \frac{4}{3} \right)^x \) involves understanding how these functions behave across different values of \( x \).

To plot the graph, we first choose a range of \( x \)-values. For each \( x \), calculate the corresponding \( y \)-value using the exponential function. This creates coordinate points \( (x, y) \) that we can plot on the graph.

• Start with a set of simple \( x \)-values such as -2, -1, 0, 1, 2.
• Substitute each value into the function to get \( y \).
• Plot these points on a Cartesian plane.

After plotting, draw a smooth curve through these points. In exponential growth, the curve will go upwards steeply as \( x \) increases. The curve becomes nearly flat to the left, approaching the \( x \)-axis but never touching it. This is indicative of what happens when the base of the exponential function is greater than 1.

Function Analysis: Insights and Interpretations

Function analysis is all about understanding the behavior and characteristics of a function. For \( y = \left( \frac{4}{3} \right)^x \), recognizing distinct properties of the function is crucial.

Since we have identified this function as an exponential growth, let's consider what other insights we can derive:

• The function has a horizontal asymptote at \( y = 0 \). This means as \( x \) approaches negative infinity, \( y \) gets closer to 0 but never actually reaches it.
• The graph will always be above the x-axis (\( y > 0 \)), reflecting positive outputs for all \( x \)-values.
• At \( x = 0 \), \( y \) equals 1 because any non-zero number raised to the power of 0 is 1.

Analyzing these components helps highlight how exponential growth functions differ from linear or other types of growth, showing the dramatic rise as \( x \) increases. This interpretive skill is vital for making meaningful conclusions from mathematical models in fields such as finance, natural sciences, and technology.