Problem 25 - Algebra 2

Question

Graph each function. $$ y=2(0.5)^{x} $$

Step-by-Step Solution

Answer

The function \(y=2(0.5)^{x}\) is an exponential function that decreases as x increases; and as x tends to infinity, the function tends to 0. The graph starts at the y-intercept (0,2) and passes through points like (1,1) and (2,0.5).

Step 1: Identify the y-intercept

The y-intercept is the value of y when x = 0. So, put \(x=0\) in the function \(y=2(0.5)^{x}\) which becomes \(y=2(0.5)^{0} = 2\). So the y-intercept is at point (0,2).

Step 2: Identify additional points

Choose some values for x and calculate the corresponding values for y to understand the shape of the graph. For example when x = 1, \(y=2(0.5)^{1} = 1\), and when x = 2, \(y=2(0.5)^{2} = 0.5\).

Step 3: Draw the graph

Plot the y-intercept and the points from step 2 on a graph. The graph will start at y = 2 when x = 0, then decrease as x increases because base of the exponent (\(0.5\)) is between 0 and 1. As x tends to infinity, the function tends to 0, but it will never become exactly zero. The graph passes through points (0, 2), (1, 1), and (2, 0.5) among others.

Key Concepts

Graphing Exponential Functions

Graphing an exponential function involves plotting a curve that rises or falls rapidly. These functions have the form \(y = ab^x\), where \(a\) is the initial value and \(b\) is the base of the exponent. To graph an exponential function, follow these steps:

• Identify the initial value \(a\), which tells you the starting point when \(x = 0\). This is also the y-intercept of the graph.
• Determine the base \(b\) to see how the function will grow or decay. If \(b > 1\), the function exhibits exponential growth. If \(0 < b < 1\), it displays exponential decay.
• Select a few values of \(x\) and calculate corresponding \(y\) values to get additional points which help define the curve's shape.

These ingredients allow us to create a smooth curve that efficiently represents the function's behavior. With exponential functions, the change is compounded, meaning small increases in \(x\) can lead to large changes in \(y\). This makes the graphing particularly intriguing as it displays a distinctive bending curve.

Exponential Decay

Exponential decay occurs when the base of the exponential function is between 0 and 1, like \(b = 0.5\) in the function \(y = 2(0.5)^x\). This base causes the function's value to decrease rapidly as \(x\) increases.

• Each step rightward along the \(x\)-axis results in halving (or reducing by a consistent factor) of the \(y\)-value.
• This characteristic creates a curve that approaches zero but never actually reaches it; the curve asymptotes towards the x-axis.
• Real-world examples include radioactive decay and cooling of a hot object, where the decrease is rapid initially and slows over time.

Exponential decay is fascinating because it visually shows how processes slow down over time as the rate of decrease also diminishes. It's an intuitive way to understand the concept of diminishing returns or decrease rates.

Y-intercept of Exponential Functions

The y-intercept of an exponential function is a crucial point on the graph. It's where the graph crosses the y-axis, occurring when \(x = 0\). To find the y-intercept of the function \(y = 2(0.5)^x\), substitute \(x = 0\) into the function:

• Calculating \(y = 2(0.5)^0\), we find \(y = 2\) because any number raised to the power of 0 is 1.
• This gives the point \((0, 2)\), ensuring an easily identifiable starting point on the graph.
• The y-intercept represents our initial value or condition in many real-world problems.

Understanding the y-intercept is essential as it provides insights into the initial state or starting condition of an exponential process. It's a vital stepping stone for sketching the graph and interpreting real-life applications where the initial measurement is crucial.