Problem 9 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=-2^{x} $$

Step-by-Step Solution

Answer

The graph of \(y = -2^x\) is a reflection of \(y = 2^x\) over the x-axis, decreasing rapidly to the right.

Step 1: Identify the Base Function

The given function is \(y = -2^x\). The base function is \(y = 2^x\), which is an exponential function with base 2. The exponential function \(2^x\) has a domain of all real numbers and a range of positive real numbers.

Step 2: Recognize the Transformation

The given function is \(y = -2^x\). The negative sign in front of \(2^x\) reflects the graph of \(2^x\) over the x-axis. This means that for every point \((x, 2^x)\) on the graph of \(y = 2^x\), the corresponding point on the graph of \(y = -2^x\) will be \((x, -2^x)\).

Step 3: Create a Table of Values

Choose a few values of \(x\) to find corresponding \(y\) values. For example, calculate:- \(x = -2\), \(y = -2^{-2} = -\frac{1}{4}\)- \(x = -1\), \(y = -2^{-1} = -\frac{1}{2}\)- \(x = 0\), \(y = -2^0 = -1\)- \(x = 1\), \(y = -2^1 = -2\)- \(x = 2\), \(y = -2^2 = -4\)

Step 4: Plot the Points and Draw the Graph

Plot the calculated points on a coordinate plane: \((-2, -\frac{1}{4})\), \((-1, -\frac{1}{2})\), \((0, -1)\), \((1, -2)\), \((2, -4)\). Connect these points smoothly with a curve.

Step 5: Describe the Graph Behavior

The graph of \(y = -2^x\) approaches 0 as \(x\) approaches negative infinity, indicating a horizontal asymptote at \(y = 0\). As \(x\) increases, \(y\) decreases rapidly, and the graph moves downward from left to right.

Key Concepts

Graphing Transformations

When we deal with exponential functions like \[y = 2^x\], we often need to learn how to transform these graphs, making them look different on the coordinate plane. In particular, any transformations applied to the function alter its shape and position in notable ways. Let's start with one of the more straightforward transformations—the reflection over the x-axis. The given function is \[y = -2^x\], which can be compared to the base function \[y = 2^x\]. The negative sign in front plays a significant role here. It flips the entire graph upside down. This means that every y-value of the base function becomes its negative in the transformed one:

• If \[(x, y)\] is a point on \[2^x\], then \[(x, -y)\] is the corresponding point on \[-2^x\].

This reflection causes the graph to move downward rather than upward as x increases. It's an essential concept, showing how the manipulation of a function's equation directly impacts the visual representation of the graph.

Horizontal Asymptote

A key feature of exponential functions is the horizontal asymptote, which is a line that the graph approaches but never quite touches or crosses. For the function \[y = -2^x\], the horizontal asymptote is at \[y = 0\]. This means that as \[x\] becomes more and more negative (moves towards negative infinity), the value of \[y\] gets closer and closer to zero.The role of a horizontal asymptote is to provide a boundary for the graph's behavior in one direction. Even for a downward-facing graph like this one, as it stretches further to the left, the values hover near zero. Understanding asymptotes helps us predict the long-term behavior of the graph:

• As \[x \to -\infty\], \[y \to 0^-\] (approaches zero from the negative side).

In practice, recognizing this boundary helps us sketch more accurate graphs and solve many practical applications concerning limits and trends in functions.

Domain and Range

The domain and range of exponential functions reveal the limits within which a function exists concerning x and y values respectively. For the function \[y = 2^x\], the domain encompasses all real numbers because \[x\] can be any real number.When reflecting across the x-axis to form \[y = -2^x\], the domain remains unchanged:

• **Domain:** All real numbers.

This characteristic tells us that no matter how the graph is transformed, \[x\] can have any value, whether positive, negative, or zero.The range, on the other hand, reflects the effect of the transformation. Whereas \[y = 2^x\] had a range of positive real numbers, \[y = -2^x\] has a range of negative real numbers, including zero:

• **Range:** Negative real numbers and zero.

Understanding these concepts is crucial when exploring other transformation effects and paves the way for more advanced mathematical insights.