Problem 31 - Algebra 2

Question

Graph each function. $$ y=\frac{1}{2}\left(\frac{1}{2}\right)^{x} $$

Step-by-Step Solution

Answer

The function \(y=\frac{1}{2}\left(\frac{1}{2}\right)^{x}\) is a decreasing exponential function with a y-intercept of \(1/2\) and an horizontal asymptote at \(y = 0\). Some key points on the graph are \((-2, 4)\), \((-1, 2)\), \((0, 1)\), \((1, 1/2)\), and \((2, 1/4)\).

Step 1: Identify the Function's Properties

Firstly, recognise this is an exponential function of the form \(y=ab^{x}\), where \(a\) is the initial value, and \(b\) is the base. The base is \(\frac{1}{2}\), which makes the function a decreasing exponential function. The initial value (a) is \(\frac{1}{2}\), which is the y-intercept of the function.

Step 2: Find Some Key Points

To have a better view of the function, calculate the output of the function at key points. For instance, when \(x = -2\), \(x = -1\), \(x = 0\), \(x = 1\), and \(x = 2\), the function outputs \(4\), \(2\), \(1\), \(1/2\), and \(1/4\), respectively.

Step 3: Construct the Graph

Mark the y-intercept, which is \(1/2\), and plot the points calculated. Then, draw a curve passing through these points. As \(x\) goes to infinity, the function approaches \(0\) (the x-axis), but it will never actually reach it. This results in the horizontal asymptote \(y = 0\).

Key Concepts

Exponential Decay

Exponential decay occurs in a function when the quantity decreases at a constant proportional rate.
In simple terms, this means that as the variable (usually time, denoted as \(x\)) increases, the value of the function decreases. The rate of decay can be identified by the base of the exponential function.

• In our exercise, the function is given as \(y = \frac{1}{2}\left(\frac{1}{2}\right)^{x}\).
• The base is \(\frac{1}{2}\). When the base is between 0 and 1, the function represents exponential decay.

This means the function's output gets smaller as \(x\) increases, demonstrating a drop in value over time. A practical example of exponential decay is radioactive decay, where substances decrease in amount over a given time frame.

Function Graphing

Graphing an exponential function like \(y = \frac{1}{2}\left(\frac{1}{2}\right)^{x}\) helps us visualize how the function behaves across different values of \(x\).
Initial steps for graphing involve determining key points and understanding the curve's general direction.

• Start by calculating the y-values for a selection of x-values (e.g., \(x=-2, -1, 0, 1, 2\)). For our function, these correspond to \(y\) values of 4, 2, 1, 0.5, and 0.25 respectively.
• Plotting these points on a graph gives an idea of the decreasing trend due to exponential decay.

Connect these points smoothly to create a curve, noting the slope and direction of the graph. The curve should slope downward left to right, getting closer to the horizontal axis but never touching it.

Horizontal Asymptote

A horizontal asymptote in the graph of an exponential function is a horizontal line that the graph approaches but never actually reaches.
It gives us an understanding of the behavior of the function as \(x\) tends towards infinity.

• In our case, the horizontal asymptote is the line \(y = 0\), which is the x-axis.
• As \(x\) increases without bounds, the value of \(y\) approaches zero but never becomes zero.

This behavior shows that even with large changes in \(x\), the function’s value will decrease but not become exactly zero. Understanding the concept of horizontal asymptotes is crucial in predicting the long-term behavior of the function.

Initial Value

The initial value of an exponential function is the starting value, which occurs when \(x = 0\). It is denoted by the parameter \(a\) in the general exponential form \(y = ab^x\).
This initial value is crucial as it determines the y-intercept in the graph.

• For our function \(y = \frac{1}{2}\left(\frac{1}{2}\right)^{x}\), the initial value \(a\) is \(\frac{1}{2}\).
• This means that when \(x = 0\), the function value \(y\) is \(\frac{1}{2}\).

The initial value gives a benchmark for how the function starts before any changes due to \(x\) occur. It is also the point where the graph intersects the y-axis, making it a vital reference for drawing the function's graph.