Problem 26 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph the function. $$ y=-5^{x} $$

Step-by-Step Solution

Answer

To graph the function \( y = -5^x \), you understand that this is an exponential function reflected in the x-axis. Using a graphing utility, you can easily plot this function and see its characteristics visually.

Step 1: Understand the Basic Form of the Exponential Function

First, it's crucial to understand the basic form of an exponential function, which is \( y = a^x \) where \( a > 0, a \neq 1 \). For \( a > 1 \), as \( x \) increases, \( y \) also increases and as \( x \) decreases, \( y \) approaches 0 but is always positive. The graph passes through the point (0, 1) because any non-zero number raised to the power of 0 equals 1.

Step 2: Understand the Impact of the Negative Sign

The negative sign in \( y = -5^x \) will reflect the basic exponential function \( y = 5^x \) in the x-axis. This means that for \( x \) less than, equal to, or greater than 0, \( y \) is respectively greater than, equal to, or less than 0.

Step 3: Plot the Function Using a Graphing Utility

Now the function can be plotted using a graphing utility. Just go to the graphing utility, input the function \( y = -5^x \) and draw the graph. This will give a clear visual representation of the function. You will see that the graph is an reflection of the function \( y = 5^x \) in the x-axis.

Key Concepts

Graphing Utilities

Graphing utilities are tools that help visualize mathematical functions on a coordinate plane. By inputting an equation, like the one given in the exercise, these tools can quickly plot the corresponding graph, providing a clear visual representation. They are incredibly useful for exponential functions, such as \( y = -5^x \), because they can instantly show the behavior and shape of the graph. Using a graphing utility, like a scientific calculator or a computer application, saves time and can help check the accuracy of manually drawn graphs. To graph \( y = -5^x \), simply input the equation into the utility. You'll see a downward-sloping curve, as the utility reflects the exponential growth negatively on the y-axis.

• Start by accessing your graphing tool.
• Input the function \( y = -5^x \).
• Observe the reflected curve and note how it behaves as \( x \) changes.

By using graphing utilities, you can explore various functions and understand their properties more effectively.

Negative Exponents

Negative exponents represent reciprocal powers. For instance, \( a^{-x} \) is equal to \( \frac{1}{a^x} \). In the context of the given function \( y = -5^x \), the exponent is positive, but the negative sign outside indicates a reflection, not a negative exponent per se.When dealing with negative exponents in general:

• Understand that \( 2^{-3} \) means \( \frac{1}{2^3} = \frac{1}{8} \).
• Negative exponents result in fractions, or numbers between 0 and 1, for \( x > 0 \).

The notion of negative exponents helps to explore how functions behave when decreasing, providing insights into division and fraction-based transformations in exponential graphs.

Reflections on the Axis

Reflections alter the graph of a function, creating a mirror image over a specific axis. In the function \( y = -5^x \), the negative sign causes a reflection over the x-axis. This flip results in every y-value of the parent function \( y = 5^x \) becoming its opposite.Here's how axis reflections affect graphs:

• A negative sign before an exponential term flips the graph over the x-axis.
• Reflecting over the y-axis would require negating the exponent, leading to \( y = 5^{-x} \).

This reflective property highlights how simple modifications in the function's equation, like adding a negative sign, can significantly change the graph's appearance and behavior, offering an essential tool for developing an intuitive understanding of functions.