Problem 20 - Algebra 2

Question

Sketch the graph of each function. Then state the function's domain and range. $$ y=0.5(4)^{x} $$

Step-by-Step Solution

Answer

Domain: (-∞, +∞), Range: (0, +∞)

Step 1: Identify the Function Type

The given function is an exponential function in the form of \( y = a(b)^x \) where \( a = 0.5 \) and \( b = 4 \). This means that the function will exhibit exponential growth.

Step 2: Determine the Domain

For an exponential function \( y = a(b)^x \), the domain is all real numbers because you can substitute any real number for \( x \) without restriction. Thus, the domain is \( (-\infty, +\infty) \).

Step 3: Determine the Range

Since \( b > 1 \), the exponential function will always output positive values for \( y \). Therefore, the graph approaches the x-axis but never touches it. This means the range is \( (0, +\infty) \).

Step 4: Plot Key Points

To sketch the graph, you can plot some key points. For example:- When \( x = 0 \), \( y = 0.5(4)^0 = 0.5 \)- When \( x = 1 \), \( y = 0.5(4)^1 = 2 \)- When \( x = -1 \), \( y = 0.5(4)^{-1} = 0.125 \)Connect these points smoothly, understanding that the curve will rise rapidly as \( x \) increases and approach zero as \( x \) decreases.

Step 5: Sketch the Graph

Using the calculated key points, sketch the curve starting below the y-axis when \( x \) is negative, crossing at \( (0, 0.5) \), and continuing upward as \( x \) increases. The curve never touches the x-axis, emphasizing the positive range.

Step 6: Summarize Domain and Range

Finally, summarize the domain and range:- Domain: \( (-\infty, +\infty) \)- Range: \( (0, +\infty) \)

Key Concepts

Graphing Functions

Graphing functions can be an exciting way to visualize how different mathematical rules unfold on a plane. In the case of our original exercise, we're dealing with an exponential function. This particular type of function is in the form of \( y = a(b)^x \). When graphing, it's helpful to recognize that these graphs have a consistent pattern depending on the value of \( b \).
In our function, \( a = 0.5 \) and \( b = 4 \). Begin by identifying some easy-to-calculate points:

• When \( x = 0 \), \( y = 0.5(4)^0 = 0.5 \)
• When \( x = 1 \), \( y = 0.5(4)^1 = 2 \)
• When \( x = -1 \), \( y = 0.5(4)^{-1} = 0.125 \)

Use these points to guide your graph. Begin plotting these simple points on your graph paper. As \( x \) increases, notice the steep upward swing of the curve. Conversely, as \( x \) decreases, the curve approaches, but never touches, the x-axis. Remember: exponential functions like these grow pretty quickly!

Domain and Range

Understanding the domain and range of functions is crucial when sketching their graphs. For exponential functions such as \( y = a(b)^x \), the domain and range have unique characteristics. Let's break it down.
The **domain** of this function is straightforward: it includes all real numbers. You can substitute any real number into \( x \) without any constraints. Thus, the domain is:

• Domain: \( (-\infty, +\infty) \)

Next, let's talk about the **range**. Because \( b \) is greater than 1 in our function, it ensures that the outcome of the function \( y \), remains positive no matter which real number we plug in for \( x \). Hence, the function will only approach zero and never become negative, giving it the range:

• Range: \( (0, +\infty) \)

Also, keep in mind the behavior of the graph: it gets infinitely closer to zero but never actually touches the x-axis. This hints at the infinite nature of exponential functions.

Exponential Growth

Exponential growth is a fascinating phenomenon where things grow at a constantly increasing rate. Our function \( y = 0.5(4)^x \) is a perfect illustration of this concept. Here's why.
In an exponential function where \( b > 1 \), as is the case here with \( b = 4 \), we witness rapid growth as \( x \) increases. Every increment of \( x \) results in a growth factor of four for \( y \), thereby explaining the curve's steep ascent on the graph.
This characteristic can be seen in real-world events such as populations, interest rates, and even certain technologies where output keeps doubling or quadrupling over set intervals. In mathematics, exponential growth helps us predict and understand these phenomena. It’s pivotal to grasp the concept because it appears often, with applications that span across various fields.
Remember, exponential growth isn’t linear—it's all about powers. With every unit increase in \( x \), the value of the function skyrockets, which is evident on the graph.