Problem 62 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The intersection points of the graphs of the equations \(y=\sqrt{x}\) and \(y=x\) are (0,0) and (1,1).

Step 1: Graph the equations

Plot both \(y=\sqrt{x}\) and \(y=x\) on a graph to visualize their intersection. Notice the points where both the curves meet. Let's say the point of intersection is (a,b)

Step 2: Set the equations equal to each other

Set \(\sqrt{x}=x\) This is because at the intersection, the y-values are the same.

Step 3: Solve for x

When we square both sides, we get \(x^2=x\). Subtract 'x' from both sides to get \(x^2-x=0\). Factorize the equation to get \(x(x-1)=0\). Setting each factor to zero gives \(x=0\) and \(x=1\)

Step 4: Substitute the x-values

Substitute each x-value into the original equations to see if they hold true. Substituting x=0 and x=1 into \(y=\sqrt{x}\) and \(y=x\) yields y=0 and y=1 respectively, which validate our solutions.

Key Concepts

Graphing Utility

A graphing utility is a type of software or electronic calculator that is used to display graphs of equations, often in real-time. This tool is invaluable when attempting to find points of intersection between different functions, such as a square root and a linear function.

When using a graphing utility:

• Input the equation of each function. In our exercise, these are the square root function \(y=\sqrt{x}\) and the linear function \(y=x\).
• Observe the plotted curves and look for points where they intersect. These points are crucial to solving the problem.
• Zoom in on intersections if needed, to precisely identify the coordinates.

Utilizing a graphing utility not only helps you visually understand the relationship between two functions but also simplifies the confirmation of algebraic solutions.

Algebraic Solution

Finding an algebraic solution complements the visual approach provided by graphing utilities. Once possible intersection points are visually identified, an algebraic method verifies these results.

Follow these steps:

• Set the two functions equal: \(\sqrt{x} = x\). At the intersection, both y-values for the functions are the same.
• Square both sides to simplify: turning \(\sqrt{x} = x\) into \(x^2 = x\).
• Rearrange the equation to find roots: \(x^2 - x = 0\) becomes \(x(x - 1)=0\).
• Solve for \(x\): This implies \(x = 0\) or \(x = 1\).

These values for \(x\) provide possible intersection points that you can verify by plugging back into the original equations.

Square Root Function

The square root function \(y = \sqrt{x}\) is a basic yet crucial mathematical concept representing the principal (or positive) square root of \(x\). Its graph is a curve that starts at the origin and increases slowly to the right. Some key properties include:

• The domain is \(x \geq 0\), as you cannot find the square root of a negative number in the set of real numbers.
• The range is also \(y \geq 0\) for real numbers, as square roots yield non-negative results.
• The graph aggressively rises initially from \(x = 0\) due to the steepness at smaller values, then flattens out as \(x\) increases.

Understanding the features and behavior of a square root function aids in predicting the nature of its graph and identifying intersections with other functions on a coordinate plane.

Linear Function

A linear function, expressed in the form \(y = mx + b\), is one of the simplest types of equations, characterized by its straight-line graph. In our context, \(y = x\) is a special case of a linear function where the slope \(m\) is 1, and the y-intercept \(b\) is 0.

Important aspects of linear functions:

• The graph is a straight line, maintaining a constant slope.
• In the specific equation \(y = x\):
  • The slope is 1, indicating a 45-degree angle line with respect to the x-axis.
  • The line passes through the origin (0,0).

Because a linear function increases (or decreases) consistently, it provides a benchmark against which other functions like a square root function can be analyzed, especially when finding points of intersection.