Problem 18 - Calculus Single Variable

Question

Plot several points, and sketch the graph of the function defined by the given expression. $$ \sqrt{x-2} $$

Step-by-Step Solution

Answer

The graph of \( \sqrt{x-2} \) starts at \( (2,0) \) and gradually increases, forming a curve.

Step 1: Identify the Domain of the Function

The function is given by \( f(x) = \sqrt{x-2} \). The square root function is defined only for non-negative arguments. Therefore, we need \( x-2 \geq 0 \), which simplifies to \( x \geq 2 \). Thus, the domain of the function is \( x \geq 2 \).

Step 2: Choose Appropriate Values of x

To plot the function, choose values of \( x \) within the domain \( x \geq 2 \). For simplicity, let's choose integers starting from 2: \( x = 2, 3, 4, 5, \) and so on.

Step 3: Calculate Corresponding y-values

Substitute each x-value into the function to find the corresponding \( y \)-value. For \( x = 2 \), \( y = \sqrt{2-2} = 0 \). For \( x = 3 \), \( y = \sqrt{3-2} = 1 \). For \( x = 4 \), \( y = \sqrt{4-2} = \sqrt{2} \). For \( x = 5 \), \( y = \sqrt{5-2} = \sqrt{3} \).

Step 4: Plot the Points

Plot the calculated points \((2,0), (3,1), (4,\sqrt{2}), (5,\sqrt{3})\) on a coordinate plane.

Step 5: Sketch the Graph

Draw a smooth curve through the plotted points, extending the curve to the right as \( x \) increases, since the domain is \( x \geq 2 \) and the function is defined for all such \( x \). Remember that the curve starts at \((2,0)\) and gradually increases, reflecting the square root function's property of increasing less quickly as \( x \) increases.

Key Concepts

Domain of a Function

When discussing functions, understanding the domain is crucial. The domain of a function refers to all the possible input values (usually x-values) that the function can accept without resulting in any undefined or illegal operations. For example, in the function given by \( f(x) = \sqrt{x-2} \), the expression under the square root symbol must be non-negative. This is because you cannot take the square root of a negative number within the set of real numbers. Therefore, we set the inequality:

• \( x - 2 \geq 0 \)
• Solving this inequality gives \( x \geq 2 \)

This tells us that the domain of the function \( f(x) \) includes all real numbers starting from 2 and going to infinity. By identifying the domain, we know that any value of \( x \) less than 2 will not satisfy the function and should not be used in plotting.

Square Root Function

The square root function is an important mathematical concept. Typically written as \( f(x) = \sqrt{x} \), this function represents the principal (or positive) square root of the input x. One of its key characteristics is that it gradually becomes less steep as x increases.Consider the transformation in our exercise, \( f(x) = \sqrt{x-2} \). Here, the graph of the square root function is shifted horizontally. The "-2" inside the square root function indicates a shift to the right by 2 units on the x-axis. Thus, the graph doesn't start at zero, but at the point where \( x = 2 \). From that point forward, the function smoothly increases.This behavior makes the square root function useful in many real-world problems where negative outputs don't make sense. It inherently starts at zero and only gives positive results, modeling scenarios where quantities can't be negative.

Plotting Points on a Graph

Plotting points is a foundational skill in graphing functions. The process begins with selecting values from the function's domain. For \( f(x) = \sqrt{x-2} \), we choose x-values such as 2, 3, 4, and 5. These values are within our domain, which is \( x \geq 2 \). We then calculate the corresponding y-values using the function:

• At \( x = 2 \), \( y = \sqrt{2-2} = 0 \)
• At \( x = 3 \), \( y = \sqrt{3-2} = 1 \)
• At \( x = 4 \), \( y = \sqrt{4-2} = \sqrt{2} \)
• At \( x = 5 \), \( y = \sqrt{5-2} = \sqrt{3} \)

Next, plot these points on the graph. Place each point where its continuous visuals meet on the x-axis and y-axis. It is essential to plot these points accurately as they form the backbone of the function's graph. After plotting, draw a smooth curve that connects all points to complete the function's sketch.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves that represent functions. It is composed of two axes that intersect at a right angle. These two axes divide the plane into four quadrants:

• The horizontal axis is called the x-axis
• The vertical axis is called the y-axis

In graphing functions, the coordinate plane serves as the stage upon which the drama of mathematics unfolds. Each point on the plane is described by a pair of numbers \((x, y)\). The x-value describes the position along the horizontal axis, and the y-value describes the position along the vertical axis.Using a coordinate system allows us to visually interpret and understand the behavior of functions. When we sketched our function \( f(x) = \sqrt{x-2} \), it became apparent how the function behaves and how it transitions smoothly across the domain. This visual representation provided by the coordinate plane makes complex functions easier to comprehend.