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A square root function is a type of function that involves the square root of a variable, often written as \(y = \sqrt{x}\). The general shape of its graph is a curve that starts at the origin (0,0) and increases slowly, getting steeper as it moves to the right.
In our exercise, we have two specific square root functions: \(y = \sqrt{x}\) and \(y = 3\sqrt{x}\).
The second function, \(y = 3\sqrt{x}\), is simply the first function multiplied by a coefficient of 3. This means it will stretch vertically by a factor of 3 compared to the original function.

• When dealing with square root functions, it's essential always to remember that the domain (possible x-values) is only non-negative numbers (x ≥ 0). This is because the square root of a negative number is not defined in the real number system.
• Similarly, the range (possible y-values) of the basic square root function is also non-negative (y ≥ 0).
• Each point on the curve of the square root function corresponds to its x-value paired with its square root as the y-value. For example, if x = 4, then y = \sqrt{4} = 2\.

Function graphing involves plotting points on a coordinate plane to visualize how the function behaves.
To graph the functions given in the exercise, we start by identifying a few key points for each function. These points are where the function outputs significant or easy-to-calculate values.
For \(y = \sqrt{x}\), some key points are:

• (0,0) because \sqrt{0} = 0\
• (1,1) because \sqrt{1} = 1\
• (4,2) because \sqrt{4} = 2\
• (9,3) because \sqrt{9} = 3\

Similarly, for \(y = 3\sqrt{x}\), we scale the y-values by a factor of 3:

• (0,0) because 3\sqrt{0} = 0\
• (1,3) because 3\sqrt{1} = 3\
• (4,6) because 3\sqrt{4} = 6\
• (9,9) because 3\sqrt{9} = 9\

With these points, you can plot them on your coordinate plane and then smoothly connect them to form the curve. Remember, the curve should reflect the increasing nature of square root functions, and for \(y = 3\sqrt{x}\), the curve will be steeper compared to \(y = \sqrt{x}\).

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves.
It consists of two perpendicular axes:

• The x-axis, which runs horizontally
• The y-axis, which runs vertically

The point where these axes intersect is called the origin, represented by (0,0).
When graphing a function, we first draw the x-axis and y-axis. For our exercise, we need to ensure our axes are labeled correctly to include all points we plan to plot.
For instance, when graphing \(y = \sqrt{x}\), we include the points (0,0), (1,1), (4,2), and (9,3). For \(y = 3\sqrt{x}\), we include (0,0), (1,3), (4,6), and (9,9).

• Start by plotting each point from the two functions on the coordinate plane.
• After plotting the points, draw smooth curves through them to form the graph of each function.
• Ensure that both graphs share the same coordinate plane for easy comparison and analysis.

This way, you can clearly see how the function \(y = 3\sqrt{x}\) stretches vertically compared to \(y = \sqrt{x}\).