Problem 31 Graph each function. $$ y=\f... [FREE SOLUTION]

Chapter 8: Problem 31

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

Short Answer

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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 by step solution

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.

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.

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 $y = 0$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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91影视

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

Short Answer

Step by step solution

Identify the Function's Properties

Find Some Key Points

Construct the Graph

Key Concepts

Exponential Decay

Function Graphing

Horizontal Asymptote

Initial Value

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