Problem 11 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ g(x)=\sqrt{x+4} $$

Step-by-Step Solution

Answer

The graph of \(g(x)=\sqrt{x+4}\) starts at \((-4,0)\) and gradually curves upward to the right.

Step 1: Understand the Domain of the Function

The function is defined where the expression under the square root is non-negative. Therefore, we have the inequality \(x + 4 \geq 0\). Solving gives \(x \geq -4\). This is the domain of the function.

Step 2: Choose Values for Table

Select values for \(x\) that are within the domain \(x \geq -4\). A suitable range could be \(-4, -3, -2, -1, 0, 1, 2\). These values cover a wide range and include the point where the function starts.

Step 3: Calculate Values for \(g(x)\)

Calculate \(g(x)\) for each value of \(x\): \(-4: \ g(-4) = \sqrt{-4+4} = \sqrt{0} = 0\) \(-3: \ g(-3) = \sqrt{-3+4} = \sqrt{1} = 1\) \(-2: \ g(-2) = \sqrt{-2+4} = \sqrt{2} \approx 1.41\) \(-1: \ g(-1) = \sqrt{-1+4} = \sqrt{3} \approx 1.73\) \(0: \ g(0) = \sqrt{0+4} = \sqrt{4} = 2\) \(1: \ g(1) = \sqrt{1+4} = \sqrt{5} \approx 2.24\) \(2: \ g(2) = \sqrt{2+4} = \sqrt{6} \approx 2.45\)

Step 4: Create a Table of Values

Organize the calculated points into a table for easier plotting: \[\begin{array}{c|c} x & g(x) \ \hline -4 & 0 \ -3 & 1 \ -2 & 1.41 \ -1 & 1.73 \ 0 & 2 \ 1 & 2.24 \ 2 & 2.45 \ \end{array}\]

Step 5: Plot the Points and Sketch the Graph

Plot the points from the table \((-4, 0), (-3, 1), (-2, 1.41), (-1, 1.73), (0, 2), (1, 2.24), (2, 2.45)\) on a coordinate plane. Connect these points smoothly to sketch the graph of \(g(x) = \sqrt{x+4}\). The curve should start at \(-4,0\) and gradually increase.

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values (or \(x\) values) for which the function is defined. This is a crucial step in understanding any function, as not all mathematical expressions work for all values. For the function \(g(x) = \sqrt{x+4}\), we need to consider when the expression inside the square root, \(x+4\), is non-negative because the square root of a negative number is not a real number in basic mathematics. To find the domain, solve the inequality \(x + 4 \geq 0\).
Subsequently, simplification leads to \(x \geq -4\). This means the domain of the function \(g(x) = \sqrt{x+4}\) is all real numbers greater than or equal to \(-4\). When sketching the graph, only use \(x\) values that adhere to this range to avoid undefined points.
Checking the domain before plotting is crucial since using valid \(x\) values guarantees that every calculated output (or \(g(x)\)) is meaningful and correct for the graph.

Square Root Function

Square root functions, like \(g(x) = \sqrt{x+4}\), involve taking the square root of their input values. These functions typically have unique characteristics:

• They start at a certain point called the radical's starting point, where the expression under the square root equals zero.
• Since the square root produces only non-negative results, the graph of such functions never dips below the x-axis.
• Square root functions are continuously increasing, meaning that as \(x\) increases, \(g(x)\) does too. This creates a curve that rises gradually.

For our function \(g(x) = \sqrt{x+4}\), the curve begins at the point \((-4, 0)\), because at \(x = -4\), \(\sqrt{-4+4} = 0\).
Then, the function increases as \(x\) becomes larger. The graph of \(g(x)\) will thus depict a smooth, gentle upward slope to the right of \(-4\). This behavior reflects the real-world applications of square root functions where gradual growth is typical, such as biological processes or resource consumption.

Table of Values

A table of values is a helpful tool for plotting functions on a graph. By calculating the output \(g(x)\) for several input \(x\) values, we can clearly see how the function behaves within its domain.
For the function \(g(x) = \sqrt{x+4}\), choose \(x\) values that accommodate the domain \(x \geq -4\). Examples include \(-4, -3, -2, -1, 0, 1, \text{and } 2\). Substituting these into the function provides us the corresponding \(g(x)\) values as follows:

• \(x = -4\), \(g(x) = 0\)
• \(x = -3\), \(g(x) = 1\)
• \(x = -2\), \(g(x) = 1.41\)
• \(x = -1\), \(g(x) = 1.73\)
• \(x = 0\), \(g(x) = 2\)
• \(x = 1\), \(g(x) = 2.24\)
• \(x = 2\), \(g(x) = 2.45\)

A table of values organizes these points, making it simpler to plot on a graph. Such organized data helps visualize the gradual increase over the domain and assists in sketching an accurate representation of the function.