Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} y &=-4 e^{-x} \\ y+3 x+8 &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The short answer will be the list of intersection points rounded to three decimal places. These results will be obtained from the graphical analysis and cross-verified by substituting them back into the equations.

Step 1: Graph the Equations

The first step is to graph the two equations using a graphing utility. The first equation is an exponential decay function that starts at \( y = -4 \) when \( x = 0 \) and approaches \( y = 0 \) as \( x \) increases. The second equation is a line with slope -3 and y-intercept 8. When the equations are plotted, intersection points will be observed.

Step 2: Approximate the intersection points

The points of intersection between the two graphs give the solutions to this system of equations. By analyzing the graph, note down the intersection points and round to 3 decimal places. Read off each intersection point from the graphing utility. There may be multiple intersection points, so ensure to capture all of them.

Step 3: Verify the intersection points

After all possible intersection points have been recorded, verification is required. Substitute each point into the original equations to check their validity. An intersection point is valid if it satisfies both the equations. If not, the point should be rejected.

Key Concepts

Exponential Decay

Exponential decay describes how a quantity decreases rapidly at first and then levels off over time. In mathematics, it is commonly expressed by an exponential function. The given equation, \(y = -4e^{-x}\), is an excellent example of exponential decay. Here:

• The negative sign indicates the function is decreasing.
• The convergence towards zero signifies decay over time.

When graphed, this curve begins at \(y = -4\) for \(x = 0\) and gently rises to approach \(y = 0\) as \(x\) increases. This visual depiction helps in understanding how the exponential decay behaves over different values of \(x\).
By plotting this equation in a graphing utility, one can visually capture this decay pattern.

Linear Equation

Linear equations are the simplest forms of equations, known for producing straight lines on a graph. The equation \(y + 3x + 8 = 0\) can be rearranged to the slope-intercept form: \(y = -3x - 8\). In this form, it is easier to recognize the components:

• The slope of the line is \(-3\), indicating a downward direction.
• The y-intercept is \(-8\), which is the point where the line crosses the y-axis.

Linear equations such as this one are foundational in graphing because they provide a straight line as a reference for determining intersections with other graphs. Graphing linear equations is straightforward, making it simple to compare with other more complex function graphs like exponential ones.

Intersection Points

Intersection points are crucial in solving systems of equations graphically. They represent the values of \(x\) and \(y\) that satisfy both equations simultaneously. For example:

• Where the exponential decay graph \(y = -4 e^{-x}\) intersects with the straight line graph \(y = -3x - 8\).

To find these points, a graphing utility is necessary. Once graphed, you can spot where the curves meet. These locations are the intersection points. Each intersection is important because it provides a potential solution to the system.
After identifying them, it's crucial to verify these solutions by plugging them back into the original equations to ensure they satisfy both. If they do, these coordinates are the correct intersection points.

Graphing Utility

Graphing utilities are tools used to plot and analyze mathematical functions graphically. They offer a visual representation which might be difficult to grasp through equations alone. By using a graphing utility for the system:

• You plot \(y = -4e^{-x}\), the exponential path.
• Overlay \(y = -3x - 8\), the linear trajectory.

The utility effectively allows users to visualize where these graphs intersect. This graphical approach is beneficial for approximating exact intersection points, seeing patterns, and understanding behaviors of functions. Users can zoom in on particular areas, change scales, and refine plots to make precise observations. This makes graphing utilities indispensable for solving systems of equations graphically, especially for students seeking to understand the concept of intersections and solutions visually.