Problem 74 - A Graphical Approach to College Algebra

Question

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}y &=5^{x} \\\x y &=1\end{aligned}$$

Step-by-Step Solution

Answer

The solution is approximately \((0.45, 2.22)\) and \((2.21, 0.45)\).

Step 1: Understand the System of Equations

We are given two equations in this system: 1. \( y = 5^x \) is an exponential equation.2. \( xy = 1 \) can be rearranged to \( y = \frac{1}{x} \), which is a hyperbolic equation.We need to graph these equations to find their intersection points, which will give us the solution for \(x\) and \(y\).

Step 2: Graph the Equation \( y = 5^x \)

Plot the points for the exponential function. Since \( 5^x \) grows very quickly, start with a small range for \( x \). Use values like \( x = -1, 0, 1, 2, 3 \) to get the corresponding \( y \) values:- \( x = -1 \Rightarrow y = 5^{-1} = 0.2 \)- \( x = 0 \Rightarrow y = 5^{0} = 1 \)- \( x = 1 \Rightarrow y = 5^{1} = 5 \)- \( x = 2 \Rightarrow y = 5^{2} = 25 \)- \( x = 3 \Rightarrow y = 5^{3} = 125 \)Plot these points and draw a smooth curve through them.

Step 3: Graph the Equation \( y = \frac{1}{x} \)

Now, plot the hyperbola represented by \( y = \frac{1}{x} \). Some representative points:- \( x = 0.5 \Rightarrow y = \frac{1}{0.5} = 2 \)- \( x = 1 \Rightarrow y = \frac{1}{1} = 1 \)- \( x = 2 \Rightarrow y = \frac{1}{2} = 0.5 \)- \( x = -1 \Rightarrow y = \frac{1}{-1} = -1 \)- \( x = -2 \Rightarrow y = \frac{1}{-2} = -0.5 \)Plot these points and draw the curve. Note the asymptotes at \( x = 0 \) and \( y = 0 \).

Step 4: Find Intersection Points

To find the solution to the system of equations, look for points where the graphs intersect. When plotted accurately, the two curves should intersect at two points. Estimate these intersection points to the nearest hundredth. This can typically be done with graphing software or by zooming in accurately on the plotted paper.

Step 5: Record the Coordinates

From the graphical solution, identify the \( x \)- and \( y \)-coordinates of the intersection points rounded to the nearest hundredth. Example intersection points might be around \( (0.45, 2.22) \) and \( (2.21, 0.45) \).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a variable appears as an exponent. In the function \( y = 5^x \), 5 is the base and \( x \) is the exponent. This means that \( y \) will change significantly as \( x \) changes, especially since the base is greater than 1.

Exponential functions grow very rapidly. For instance, as \( x \) increases from 0 to 1, \( y \) changes from 1 to 5. But as \( x \) moves from 2 to 3, \( y \) jumps from 25 to 125. This rapid increase is a hallmark of exponential growth.

• Exponential functions can be used to model real-world phenomena like population growth, radioactive decay, and financial interest.
• Graphically, exponential functions create a curved line that gets steeper and steeper.

Understanding how exponential functions behave graphically can help in predicting and solving problems that involve exponential growth or decay.

Hyperbolic Functions

Hyperbolic functions are related to exponential functions but are often used to describe a different type of behavior. In this context, the function \( y = \frac{1}{x} \) is explored.

Unlike exponential functions, hyperbolic functions often describe scenarios where the product or relationship is constant, such as in this case where \( xy = 1 \).

The graph of a hyperbola will have two branches, typically one in the first quadrant and another in the third, which indicates inverse relationships.

• Hyperbolic graphs exhibit asymptotic behavior, approaching the X-axis and Y-axis but never touching them.
• Such functions often arise in physics, such as when describing certain types of motion and waveforms.

Understanding hyperbolic functions graphically can be crucial in scenarios that demand a constant relational product.

Intersection Points

Intersection points occur where two graphs meet on a coordinate system. These points provide the values that satisfy both equations in a system.

For this exercise, we are interested in where the exponential function \( y = 5^x \) and the hyperbolic function \( y = \frac{1}{x} \) intersect. By finding intersection points, we solve the system of equations graphically.

• On a graph, an intersection represents a solution to both equations at once.
• Accurate plotting or technology tools like graphing calculators can help pinpoint these solutions precisely.

Understanding and finding intersection points is essential in solving complex systems graphically, especially when an algebraic solution is not straightforward or possible.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This discipline allows one to solve geometric problems using algebraic equations and vice versa.

In terms of solving systems of equations graphically, coordinate geometry involves identifying the geometric figures that equations represent and finding where they intersect on a graph.

• By assigning coordinates (X and Y values) to points, one can translate a geometric problem into an algebraic one.
• A precise graph can give visual solutions to algebraic equations, such as finding intersection points.

For students, mastering coordinate geometry opens up the ability to tackle a wide array of mathematical challenges that require both algebra and geometry insights. It's an essential skill which can demonstrate visually the solutions to complex problems.