Problem 13 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=-\left(\frac{1}{4}\right)^{x} $$

Step-by-Step Solution

Answer

The graph is a downward decreasing curve reflected over the x-axis, approaching 0 but never reaching it.

Step 1: Understand the Function

We are dealing with the exponential function \( y = -\left(\frac{1}{4}\right)^{x} \). This function is similar to a basic exponential function \( y = a^x \) but with some modifications. The negative sign indicates that the graph is reflected over the x-axis.

Step 2: Identify Base Information

The base of the exponential function is \( \frac{1}{4} \). This means it is a decreasing function because the base is less than 1. The negative sign means the entire graph is flipped downward across the x-axis.

Step 3: Plot Key Points

Choose a few key values for \( x \), such as -2, -1, 0, 1, and 2, and evaluate the function \( y = -\left(\frac{1}{4}\right)^{x} \) at these points. Calculate the result for each:- For \( x = -2 \), \( y = -\left(\frac{1}{4}\right)^{-2} = -16 \)- For \( x = -1 \), \( y = -\left(\frac{1}{4}\right)^{-1} = -4 \)- For \( x = 0 \), \( y = -\left(\frac{1}{4}\right)^{0} = -1 \)- For \( x = 1 \), \( y = -\left(\frac{1}{4}\right)^{1} = -\frac{1}{4} \)- For \( x = 2 \), \( y = -\left(\frac{1}{4}\right)^{2} = -\frac{1}{16} \)

Step 4: Sketch the Graph

Using the points calculated in Step 3, plot them on a coordinate grid. You should see that the graph starts from a low value (e.g., -16), crossing closer to zero as \( x \) increases. The graph will never actually reach 0 since exponential functions have horizontal asymptotes.

Step 5: Asymptote and Behavior Analysis

Observe that the graph has a horizontal asymptote at \( y = 0 \) on the x-axis. As \( x \) tends to positive infinity, \( y \) approaches 0 from below. Conversely, as \( x \) tends to negative infinity, \( y \) becomes very negative.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions can seem complicated at first, but it's all about understanding how these functions behave. An exponential function, such as \( y = -\left(\frac{1}{4}\right)^{x} \), involves a base raised to the power of \( x \). In this equation, the function is graphed by calculating several key points. Consider using values for \( x \) like -2, -1, 0, 1, and 2 to chart your graph. When you evaluate the function, these calculations become your plot points, forming the curve of the graph.

As these points are plotted, a pattern emerges that shows how the exponential function behaves. The shape of the graph for an exponential function often looks like a curve that rapidly increases or decreases. In this case, because of the base \( \frac{1}{4} \) being less than one, and the negative sign, the graph decreases and is reflected.

Properties of Exponential Functions

Exponential functions have unique properties that distinguish them from other types of functions. One prominent feature is their rate of change. These functions change quite rapidly, exemplifying either steep growth or decay. This depends on the base being greater or less than one.

For \( y = -\left(\frac{1}{4}\right)^{x} \), the base \( \frac{1}{4} \) is less than one, indicating a decreasing pattern. The negative sign implies a reflection along the x-axis, flipping the graph downward. The fundamentals of exponential functions include:

• The base determines the growth (base > 1) or decay (base < 1).
• The exponent \( x \) dictates the direction (increasing or decreasing).
• A negative sign before the base indicates the graph is reflected over the x-axis.
• Exponential functions never touch their horizontal asymptote, creating a perpetual approach.

Horizontal Asymptotes

A key characteristic of exponential functions is their horizontal asymptote, which acts like an invisible boundary. In this case, the horizontal asymptote is at \( y = 0 \). The curve of the graph gets infinitely closer to this line but never actually touches it.

As an exponential function progresses towards infinity along the x-axis, its value trends towards this horizontal line. In \( y = -\left(\frac{1}{4}\right)^{x} \), as \( x \) increases, the function's graph approaches \( y = 0 \) from below due to the reflection caused by the negative sign. Even though it seems like it might reach zero, it never does, illustrating the perpetual approaching behavior typical of exponential functions.

Reflected Graphs

A reflection in graphing is like flipping the graph over a specific axis. For the function \( y = -\left(\frac{1}{4}\right)^{x} \), the graph is reflected over the x-axis due to the negative sign. This reflection means that whatever points you plot will display a mirror image below the x-axis.

Imagine you had no negative sign, the graph, instead of decreasing into negative values, would rise and approach the asymptote from above. When a negative is introduced, each value calculated flips to its opposite on the vertical axis. Reflections change the orientation:

• A reflection about the x-axis turns the curve upside down.
• The behavior and direction of the graph are swiftly altered.
• Reflections can influence the interpretation and intersections with axes significantly.

Understanding reflections is crucial to effectively graph and predict the behavior of functions.