Problem 53 - 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. $$\left\\{\begin{aligned} y &=e^{x} \\ x-y+1 &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution requires using a graphing tool to visually determine the points of intersection. The exact coordinates will depend on the tool and may require approximation. To verify, plug the intersection points into both original equations, they should satisfy both of them.

Step 1: Graphing the Equations

Use a graphing utility to plot both equations on the same set of axes. The equation \(y = e^x\) will create an increasing exponential curve, while the second equation, \(x - y + 1 = 0\), equivalent to \(y = x + 1\), would draw a straight line with a slope of 1 and y-intercept at 1.

Step 2: Approximate the Points of Intersection

By visually examining the graph or using the intersection tool provided by the graphing utility, approximate the coordinates of the points where the graphs intersect. These are the solutions to the system of equations. Round these coordinates to three decimal places.

Step 3: Verification

To verify your solutions, substitute the x and y-values of your approximated points of intersection into the original equations. If both equations are satisfied, then the solution is correct.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where the independent variable, often "x," appears as an exponent. The most common form is expressed as \(y = a^x\), where "a" is a constant and is known as the base of the exponential function. In many cases, the base "a" is the number "e," which is approximately 2.71828. When this constant "e" is used, the function takes the form \(y = e^x\).

Exponential functions have unique characteristics:

• Growth or Decay: Exponential functions model rapid growth or decay. Growth occurs if the base is greater than 1; decay if it's between 0 and 1.
• Increasing Curve: In an exponential growth function like \(y = e^x\), the graph increases rapidly and becomes steeper as x increases.
• Horizontal Asymptote: The x-axis (i.e., y = 0) acts as a horizontal asymptote, where the curve gets closer but never actually touches it.

Such functions are widely used in real-life applications such as population growth, radioactive decay, and financial modeling.

Linear Equations

Linear equations describe a relationship between two variables that create a straight line when graphed. The simplest and most common form is the slope-intercept form: \(y = mx + b\). Here, "m" represents the slope, indicating how steep the line is, and "b" represents the y-intercept, where the line crosses the y-axis.

Key features of linear equations include:

• Constant Slope: This means the rate of change between the variables is consistent throughout. The slope "m" depicts the change in y for each unit increase in x.
• Straight Line: Every linear equation, when graphed, forms a straight line.
• Intercepts: The graph will intersect the axes at specific points. The y-intercept is found at \(b\), and the x-intercept can be calculated by setting y to zero and solving for x.

In real-world scenarios, linear equations are crucial for understanding relationships where one quantity depends linearly on another, such as cost in relation to production.

Intersection Points

Intersection points occur where two graphs meet on a coordinate plane. In mathematical terms, these points represent the solution(s) of a system of equations. For a set of two equations, each intersection indicates a pair of (x, y) values that satisfy both equations simultaneously.

Finding the intersection points involves:

• Graphing: Use a graphing utility to plot the equations, which may be linear, exponential, or other types.
• Visual identification or use of a utility's tool: Once the graphs are plotted, look for crossing points or use the graph's intersection tool to identify these locations.
• Verification: After approximating the intersection points, substitute them back into the original equations to ensure each equation holds true.

Understanding intersection points is vital as it provides solutions to many practical problems, such as predicting equal shares, balancing equations, or finding equilibrium in economic models.