Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function. \(f(x)=\sqrt{x+3}\)

Step-by-Step Solution

Answer

The graph of the function starts at the point (-3,0), and continues upwards to the right, with the y-values increasing as the x-values increase. The graph only exists in the first and fourth quadrants as the function only produces positive y-values.

Step 1: Find the Domain of the Function

Start by determining the domain - the set of x-values - of the function. Since \(f(x)=\sqrt{x+3}\), it's clear that x+3 must be greater than or equal to zero (since the square root of negative numbers are undefined in real number system). So we solve the inequality \(x+3 \geq 0\), yielding \(x \geq -3\). The domain is therefore all real numbers greater than or equal to -3.

Step 2: Generate a Set of X-Values with their Corresponding Y-Values

Now, generate a set of x-values within the domain and calculate their corresponding y-values by plugging the x-values into the function. For instance, using -3, -2, -1, 0, 1, and 2 as x-values, you would get y-values of 0, 1, \(\sqrt{2}\), \(\sqrt{3}\), 2, \(\sqrt{5}\).

Step 3: Plot the Graph

Finally, plot the function by interpreting the x-values as points on the x-axis and the y-values as points on the y-axis. Draw a curve through the points which starts at the point (-3,0) and goes upwards to the right. Remember the graph will only exist in the first and fourth quadrants because the function only yields positive y-values.

Key Concepts

Domain of a Square Root Function

The domain is a critical aspect of any function, particularly the square root function. Understanding the domain means understanding which x-values are permissible in a function.
For the square root function, such as \(f(x) = \sqrt{x+3}\), x-values must ensure that the value inside the square root is not negative. This is because square roots of negative numbers are undefined in the real number system.
To find the domain of our function, solve the inequality \(x+3 \geq 0\). This simplifies to \(x \geq -3\), meaning the domain includes all real numbers from -3 onwards.

• Start by setting the expression inside the square root greater than or equal to zero.
• Solve the inequality to find the range of x-values.
• In this case, any x-value of -3 or higher can be used in the function f(x).

Recognizing this domain is vital as it dictates where the graph begins and extends.

Understanding Square Root Functions

Square root functions often have a unique look and behavior. The general form is \(f(x) = \sqrt{x} \), but it can have slightly different phases depending on shifts.
For instance, the function \(f(x) = \sqrt{x+3}\) includes a horizontal shift to the left by 3 units. This shifts the starting point of the graph.
Square root functions are always increasing. Once they start at a particular point, they move upwards to the right.

• The graph of \(\sqrt{x} \) begins at the origin (0,0), whereas \(\sqrt{x+3}\) starts at (-3,0).
• This shift does not change the shape, only the starting point.
• Square root functions rise slowly; their increase rate changes based on how they are shifted or stretched.

Observing and recognizing these characteristics will make it easier to predict how different forms of the function might look.

Plotting Points to Graph the Function

Plotting points for a square root function involves a straightforward process. Once we know the domain, we can select possible x-values and find the corresponding y-values.
In our example, select values like \(-3, -2, -1, 0, 1,\) and \(2\) that fit within the domain. For each x-value, substitute it into the function to find the y-value.
These x and y pairs create coordinates to plot on a graph.

• For \(x = -3\), \(y = 0\), yielding the point \((-3,0)\).
• For \(x = 0\), substituting gives \(y = \sqrt{3}\), creating the point \((0, \sqrt{3})\).
• Continue in this manner for all selected x-values.

Once all the points are plotted, connect them smoothly with a curve. The graph of \(f(x) = \sqrt{x+3}\) starts at \((-3,0)\) and trends upward to the right, growing gradually. This visual representation solidifies your understanding of how the square root function operates.