Problem 1 Graph. $$ y=4^{x} $$... [FREE SOLUTION] (2024)

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Chapter 4: Problem 1

Graph. $$ y=4^{x} $$

Short Answer

Expert verified

Plot points for multiple x-values and draw a smooth exponential curve through them.

Step by step solution

- Understanding the function

The function given is an exponential function: y = 4^xHere, the base is 4 and the exponent is x.

- Choose a range of x-values

To graph the function accurately, choose a range of values for x. A good range to start with is -2, -1, 0, 1, and 2.

- Calculate corresponding y-values

Use the function y = 4^x to compute y-values for each chosen x-value: For x = -2: y = 4^{-2} = 1/16 For x = -1: y = 4^{-1} = 1/4 For x = 0: y = 4^0 = 1 For x = 1: y = 4^1 = 4 For x = 2: y = 4^2 = 16

- Plot the points

On graph paper or using a graphing tool, plot the points you calculated: (-2, 1/16), (-1, 1/4), (0, 1), (1, 4), and (2, 16).

- Draw the curve

Draw a smooth curve through the plotted points, extending the curve through each direction to reflect the behavior of the exponential function. The curve will rapidly increase as x becomes more positive and approach zero as x becomes more negative.

- Analyze the graph

Notice that the graph is always increasing and never touches the x-axis. This behavior reflects the nature of exponential growth.

Key Concepts

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

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. For example, in the function given by this exercise, y = 4^xthe base is 4 and the variable exponent is x. The main characteristic of an exponential function is rapid growth or decay, depending on whether the base is greater than or less than one. They exhibit continuous and smooth curves and are used to model phenomena such as population growth, radioactive decay, and interest compounding.

Graph Plotting

Graph plotting is a visual way to represent data or functions. In this exercise, we plot the exponential function y = 4^xby following a few clear steps:

Select specific x-values (-2, -1, 0, 1, 2)
Calculate corresponding y-values
Plot these pairs on a coordinate grid

By connecting these points with a smooth curve, we can accurately represent the behavior of the function.
Graph plotting helps us visually analyze functions and observe important characteristics like growth rate and asymptotic behavior.

Function Behavior Analysis

Analyzing the behavior of an exponential function involves understanding how changes in the variable exponent affect the function's output. For y = 4^x:

As x increases, y grows rapidly.
As x decreases, y approaches zero but never becomes negative.

This rapid increase is typical for exponential growth. Additionally, the function never touches the x-axis, illustrating that the function's value is never zero. Observing these behaviors allows us to understand how exponential functions can model real-world exponential growth scenarios effectively.

Range and Domain

The range and domain of a function specify the possible values of x and y. For the exponential function y = 4^x:

The domain is all real numbers (-∞ < x < ∞).
The range consists of all positive real numbers (0 < y < ∞).

This means the function accepts any real number as input (x-values) and produces only positive outputs (y-values).
Understanding the domain and range is crucial for graphing functions and identifying their possible values, which helps in problem-solving across various applications.

Exponential Growth

Exponential growth describes how quantities increase rapidly over time, represented by exponential functions like y = 4^x. As x increases, the value of y grows faster, and this acceleration is characteristic of exponential functions. Examples include:

Population growth
Investment growth
Compound interest

Recognizing exponential growth patterns helps predict and analyze scenarios where change accumulates significantly over time, thus underscoring the importance of studying these functions in real-life contexts.

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$Problem 1 Graph. $$ y=4^{x} $$... [FREE SOLUTION] (3)$

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