Problem 125 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{aligned}&y_{1}=7\\\&y_{2}=2^{x-1}-5\end{aligned}$$

Step-by-Step Solution

Answer

Due to the nature of this exercise, an exact answer can only be obtained by use of a graphing utility as this answer varies by the specific tool used, however the concept described above is the general method for solving such problems.

Step 1: Graph the First Function

Using a graphing utility, first plot the graph of the function \(y_{1}=7\). This is a simple horizontal line passing through the point (0,7) on the y-axis.

Step 2: Graph the Second Function

Next, plot the function \(y_{2}=2^{x-1}-5\). This is an exponential function shifted down by 5 units. Lastly, display both plots on the same graph for comparison.

Step 3: Approximate Intersection Point

Look for the point where both graphs intersect. That would be the approximate solution to our system of equations.

Step 4: Round to Three Decimal Places

Lastly, round off your result to three decimal places. This is to ensure the answer follows the level of accuracy specified in the exercise. If the graphing utility supports it, you can set it to display the results to three decimal places thus saving you the trouble of manual rounding.

Key Concepts

Graphing Utility

When solving algebra problems, a graphing utility is an invaluable tool, especially when dealing with complex equations or systems of equations. These utilities, often available as software or online applications, allow students to visually understand and analyze mathematical concepts.

Graphing utilities can be used to plot a variety of functions by converting algebraic expressions into visual graphs. Functions like lines, parabolas, and exponentially increasing or decreasing curves are much easier to comprehend when students can see them.

Using Graphing Utilities Effectively

To effectively use a graphing utility, you should first understand the type of function you are working with, such as a constant function like \(y = 7\), which appears as a straight horizontal line. Alternatively, you might work with an exponential function, which has a distinctive curve reflecting its rate of growth or decay.

• Input the correct function into the utility
• Adjust your view window to ensure the relevant parts of the graph are visible
• Use the intersect or trace features to find the exact point where two functions meet, if possible

By taking advantage of these features, you can efficiently solve for intersections, as the graphing utility will handle the complex calculations for you.

Exponential Function

An exponential function is a unique and powerful type of mathematical function that features growth or decay at a rate that is proportional to its current value. These functions are expressed in the form \(f(x) = ab^{x}\), where \(a\) is a constant term, \(b\) is the base and it is a positive real number not equal to 1, and \(x\) is the exponent.

Exponential growth is characterized by the function rapidly increasing as \(x\) gets larger. Conversely, exponential decay happens when the function decreases as \(x\) gets larger, which typically occurs when \(0 < b < 1\).

Interpreting Exponential Functions

An important aspect of exponential functions is their graph. The exponential curve will either rise or fall rapidly, without ever touching the x-axis, illustrating concepts such as compound interest, population growth, or radioactive decay in real-life scenarios.

• Understand the properties of the exponential function
• Recognize the impact of 'shifts' and 'stretches' on the graph
• Investigate real-life examples to see how exponential functions are applied

These functions are integral to various fields and understanding how to graph and analyze them is essential for students in mathematics.

System of Equations

A system of equations is a set of two or more equations that you deal with at once. When each equation represents a different line or curve on a graph, the point of intersection between these lines represents the common solution to the system.

Systems of equations can have a single solution, no solution, or infinitely many solutions. In the case of linear systems, the lines can be parallel (no solutions), coincide (infinite solutions), or intersect at one point (a unique solution).

Finding Solutions to Systems

Solving a system of equations can be done using various methods:

• Graphing: Plot each equation on the same graph and find the point(s) where they intersect
• Substitution: Solve one equation for one variable and substitute the result into another equation
• Elimination: Add or subtract equations to eliminate a variable and solve for the other

Each method has its context and utility, and while graphing provides a visual solution, algebraic methods might be more accurate or feasible depending on the specific problem at hand.