Question

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}y=e^{x} \\ x-y+1=0\end{array}\right.$$

Short answer

The intersection point of the graphs of equations \(y = e^{x}\) and \(x - y + 1 = 0\) can initially be determined through graphing, which gives an estimate of \(x\). Solving the equation \(e^{x} = x + 1\) algebraically further confirms the exact value of \(x\) at the intersection point. The corresponding value of \(y\) is verified by substituting \(x\) into both original equations.

Step-by-step solution

Understand and plot the equations

Recognize that \(y = e^{x}\) is an exponential function where the value of \(y\) increases as \(x\) increases. The second equation after rearrangement will be \(y = x + 1\), which is a linear function. Plot these two equations using a graphing utility.

Identify the intersection point

Identify the point(s) where the two graphs intersect. This gives a general idea of the value of \(x\) for the intersection point(s).

Confirm algebraically

To confirm the intersection point(s) algebraically, set the equations equal to each other: \(e^{x} = x + 1\). Solve this equation by using numerical methods such as the Newton-Raphson method, as it's a transcendental equation and can't be solved exactly by algebraic methods.

Validate the solution

Substitute the obtained value(s) of \(x\) into both original equations to verify that they yield the same value of \(y\) and thus confirm the intersection point(s).

Key concepts

Exponential Functions

Exponential functions, as seen in the equation y = e^x, describe situations where a quantity grows at a rate proportional to its current value. In simpler terms, as x increases, y gets larger at an accelerating pace, which gives the graph of an exponential function a distinctive, rapidly rising curve.

This behavior is typical for processes such as compound interest in finance, radioactive decay in physics, and many natural growth patterns in biology. It's important for students to recognize this type of function visually and understand its behavior because exponential growth can quickly lead to very large numbers.

When graphing an exponential function, always remember:

• The y-value will never be negative, as the exponential function is always positive.
• The rate of increase becomes steeper as x increases.
• There is a horizontal asymptote on the graph which the curve approaches but never touches, commonly the x-axis.

Linear Functions

Linear functions are quite straightforward and are typically written in the form y = mx + b, where m is the slope and b is the y-intercept. The equation x - y + 1 = 0 can be rearranged to the form y = x + 1, which is a linear function with a slope of 1 and a y-intercept of 1.

The graph of a linear function is always a straight line, which makes it easier to visualize and work with. Unlike exponential functions, linear functions increase at a constant rate, represented by the slope. Understanding linear equations involves being familiar with these aspects:

• The slope, or steepness, of the line.
• The y-intercept, the point where the line crosses the y-axis.
• Any point on the line can be used to create an equation for the line.

Graphing Utility Usage

Graphing utilities, such as calculators or software applications, are invaluable tools for visualizing functions and finding their points of intersection. They allow you to quickly plot equations and analyze their behavior on a coordinate grid.

To use a graphing utility for this purpose, you input the equations of the functions you're interested in. For example, entering y = e^x and y = x + 1 separately will display their respective graphs. You can then visually identify where they cross, or use built-in features to find the intersection points precisely.

Using graphing utilities can save time and provide a clear visual representation that reveals relationships between functions, which is incredibly helpful when dealing with complex equations.

Algebraic Methods

Algebraic methods involve manipulating and solving equations using algebra. To find the intersection algebraically in the given exercise, you set the two functions equal to one another because at the intersection points, they have the same x and y values.

Equating e^x to x + 1 from our exercise, we obtain a transcendental equation which often cannot be solved using basic algebraic operations. Instead, numerical methods like the Newton-Raphson method are used to approximate solutions. These methods iteratively converge on a solution by finding successively closer approximations.

After solving for x using a numerical method, you would plug this value back into the original functions to find the corresponding y value, thus confirming the intersection point or points. Complex equations often require these advanced methods, and understanding when to apply them is crucial for solving higher-level math problems.