Problem 16 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The graph of \( y = \left( \frac{1}{2} \right)^x + 2 \) is a decreasing curve approaching \( y = 2 \) from above.

Step 1: Understand the Function

The given function is \( y = \left( \frac{1}{2} \right)^{x} + 2 \). It is an exponential function with a base of \( \frac{1}{2} \) and a vertical shift upward by 2 units.

Step 2: Identify Key Properties

This function has a horizontal asymptote at \( y = 2 \) due to the vertical shift. The base, \( \frac{1}{2} \), indicates that it is a decreasing function, because \( \frac{1}{2} < 1 \). As \( x \) increases, \( \left( \frac{1}{2} \right)^x \) approaches 0, so \( y \) approaches the asymptote.

Step 3: Create a Table of Values

Choose several values of \( x \) to create a table. For example:- \( x = -2 \): \( y = \left( \frac{1}{2} \right)^{-2} + 2 = 4 + 2 = 6 \)- \( x = -1 \): \( y = \left( \frac{1}{2} \right)^{-1} + 2 = 2 + 2 = 4 \)- \( x = 0 \): \( y = 1 + 2 = 3 \)- \( x = 1 \): \( y = \frac{1}{2} + 2 = 2.5 \)- \( x = 2 \): \( y = \frac{1}{4} + 2 = 2.25 \)

Step 4: Plot the Points

Using the table of values, plot the points \((-2, 6)\), \((-1, 4)\), \((0, 3)\), \((1, 2.5)\), and \((2, 2.25)\) on the coordinate plane.

Step 5: Draw the Graph

Connect the plotted points with a smooth curve to the right and left, approaching the horizontal asymptote \( y = 2 \) as \( x \) increases.

Key Concepts

Graphing Exponential Functions

When working with exponential functions, it's important to understand their unique characteristics. An exponential function generally takes the form \( y = a \, b^x \), where \( a \) is a constant multiplier and \( b \) is the base of the exponential. In the function \( y = \left( \frac{1}{2} \right)^x + 2 \), the base is \( \frac{1}{2} \), indicating key properties regarding the graph's shape and behavior.

To graph an exponential function like this one, follow these steps:

• Identify the base of the function. If \( b < 1 \), the function is a decreasing one, while if \( b > 1 \), it is increasing.
• Determine the vertical or horizontal shifts present in the equation. In our case, there is a vertical shift upward by 2 units.
• Create a table of values by substituting a range of \( x \) values into the equation to find corresponding \( y \) values.
• Plot these \( (x, y) \) points on a coordinate grid.
• Connect the points with a smooth curve, indicating the direction of the function as it approaches its asymptote.

By following these steps, one can effectively graph any exponential function, reflecting its distinctive curve and approaching behavior near the asymptote.

Horizontal Asymptote

Exponential functions often come with a horizontal asymptote, which is a critical concept in their behavior. A horizontal asymptote is a horizontal line that the graph of the function approaches as \( x \) moves towards positive or negative infinity, but never actually reaches in finite space.

For the function \( y = \left( \frac{1}{2} \right)^x + 2 \), the horizontal asymptote is \( y = 2 \). This is because the additional constant in the equation represents a vertical shift. Basically, it means that as \( x \) increases, \( \left( \frac{1}{2} \right)^x \) tends toward 0, thus making \( y \) get closer and closer to 2.

Understanding horizontal asymptotes helps predict the end behavior of the graph. For instance:

• In this specific function, no matter how large \( x \) becomes, the output \( y \) will never quite reach 2 but will get infinitely close.
• When graphing, it’s vital to sketch this asymptotic line to guide how the curve should be drawn as it stretches toward infinity.

Horizontal asymptotes provide a visual guideline and understanding that is imperative while graphing exponential functions.

Decreasing Functions

A decreasing function is one where the value of the function decreases as the input, or \( x \), increases. In the context of exponential functions, whether a function is increasing or decreasing depends on the base \( b \).

With the given function \( y = \left( \frac{1}{2} \right)^x + 2 \), the base \( \frac{1}{2} \) is less than 1, marking it as a decreasing function. This is because each increment in \( x \) leads to multiplying by a fraction between 0 and 1, which makes the result of \( b^x \) smaller each time.

Key points to remember about decreasing exponential functions include:

• The graph descends from left to right on the coordinate plane. As \( x \) grows, \( y \) reduces.
• Despite the decreasing aspect, the curve grows increasingly slower as it progresses towards its horizontal asymptote.
• Plotting a decreasing function involves checking that the graph approaches downward in this exponential manner.

Understanding whether an exponential function is decreasing helps ensure accurate graphing and interpretation of the function's behavior over its entire domain.