Problem 5 - Algebra 2

Question

Graph each function. \(y=\sqrt{x-3}\)

Step-by-Step Solution

Answer

The graph of the function \(y = \sqrt{x-3}\) starts at the point (3,0) and forms a gradually increasing curve for \(x > 3\). It is undefined for \(x < 3\).

Step 1: Identify the Domain

The first step in graphing any function is to identify its domain. In this case, the function \(y = \sqrt{x-3}\) is only defined for \(x \geq 3\). This is because the expression under the square root, \(x-3\), must be equal to or greater than 0.

Step 2: Plot the Points

Next, we create a table of values for \(x\) and \(y\). To do this, we choose some \(x\) values greater than or equal to 3, and calculate the corresponding \(y\) values by plugging the \(x\) values into the function. For example, let's choose \(x = 3, 4, 5, 6, 7\). Plugging these into the function, we obtain \(y = 0, 1, \sqrt{2}, \sqrt{3}, 2\). Hence the points to plot are (3,0), (4,1), (5, \(\sqrt{2}\)), (6, \(\sqrt{3}\)), and (7,2).

Step 3: Draw the Graph

Mark the points obtained in the previous step on the graph, keeping in mind the domain of the function. The graph will start at (3,0) since the point at which \(x = 3\) is the beginning of the domain of the function. The graph will form a curve that increases gradually, illustrating the square root function.

Key Concepts

Domain of a function

Understanding the domain of a function is crucial in graphing, especially for square root functions. The domain refers to all possible input values (or x-values) that make the function produce real numbers. For the square root function \(y = \sqrt{x-3}\), we focus on the expression inside the root, \(x-3\). This must be non-negative since square roots of negative numbers are not real numbers.
Therefore, we set up the inequality \(x-3 \geq 0\), which simplifies to \(x \geq 3\). Thus, the domain of this function is all real numbers greater than or equal to 3. This domain directly impacts what parts of the graph appear on our coordinate plane. Always start by finding the domain to know where your graph will begin.

Function graphing

Function graphing is the process of plotting points to visualize the mathematical relationship described by a function. To graph \(y = \sqrt{x-3}\), we start by selecting x-values from the domain you previously identified. Each x-value will correspond to a y-value, which you get by substituting x into the function:

• For \(x = 3\), \(y = \sqrt{3-3} = 0\).
• For \(x = 4\), \(y = \sqrt{4-3} = 1\).
• For \(x = 5\), \(y = \sqrt{5-3} = \sqrt{2}\).
• Continue this way to find more points.

Once you have a set of (x, y) coordinates, you plot them on a Cartesian plane. Connect these points smoothly to reflect the continuous nature of the square root function. Remember to start at \(x = 3\) as it marks the beginning of the domain.

Square root transformations

Square root transformations involve changes to the basic square root parent function \(y = \sqrt{x}\). For the function \(y = \sqrt{x-3}\), the transformation is a horizontal shift to the right by 3 units.
This is because of the term \(x-3\): it indicates that every x-value from the basic square root function is now increased by 3 to maintain the output values. As a result, the starting point (0,0) of \(y = \sqrt{x}\) shifts to (3,0).
The transformation doesn't affect the vertical shape but tells us that the graph starts at (3,0) and moves upwards to the right. Understanding transformations helps in predicting and plotting the graph accurately without extensive calculations.