Question

Graph each function. Be sure to label three points on the graph. $$f(x)=\sqrt{x}$$

Short answer

Plot the points (0, 0), (1, 1), and (4, 2). Draw a smooth curve through these points.

Step-by-step solution

- Determine Key Points

Select three key points on the graph of the function. For the function \(f(x) = \sqrt{x}\), common points to use are \((0, 0)\), \((1, 1)\), and \((4, 2)\) because they are easy to calculate and plot.

- Calculate Function Values

Plug in the values of \(x\) into the function to get the corresponding \(y\)-values. When \(x = 0\), \(f(0) = \sqrt{0} = 0\). When \(x = 1\), \(f(1) = \sqrt{1} = 1\). When \(x = 4\), \(f(4) = \sqrt{4} = 2\).

- Plot the Points

Plot the points \((0, 0)\), \((1, 1)\), and \((4, 2)\) on the coordinate plane. These points will help to sketch the graph accurately.

- Draw the Graph

After plotting the points, draw a smooth curve through these points that represents the function \(f(x) = \sqrt{x}\). This curve should be increasing and only exist in the first quadrant (values of \(x\) and \(y\) are positive or zero).

- Label the Points on the Graph

Clearly label the points \((0, 0)\), \((1, 1)\), and \((4, 2)\) on your graph so that it is clear where these key points are located.

Key concepts

Key Points

When working with functions, identifying key points is crucial. For the function given by \( f(x) = \sqrt{x} \), key points are those points which are easy to calculate and serve as reference for plotting the graph.

In this example, the points \((0, 0)\), \((1, 1)\), and \((4, 2)\) are selected because:

• They are easy to compute.
• The coordinates consist of integers, making them straightforward to plot.
• They cover different parts of the graph, providing a good representation of the function.

These key points help build a framework for sketching the graph.

Square Root Function

The square root function, represented by \( f(x) = \sqrt{x} \), is a function that pairs each non-negative number with its square root. It is important to understand the properties of this function:

• It only accepts non-negative values of \( x \) (\( x \geq 0 \)).
• The output is also non-negative (\( f(x) \geq 0 \)).
• For each visible point on its graph, both coordinates should be non-negative, placing it in the first quadrant of the coordinate plane.
• The function increases at a decreasing rate; as \( x \) increases, \( f(x) \) increases but more slowly.

Recognizing these characteristics will help in plotting the function accurately.

Coordinate Plane

The coordinate plane is essential for graphing functions, as it provides a visual map of the function's behavior. Understanding the basics of the coordinate plane includes:

• The plane is divided into four quadrants based on the signs of the coordinates.
• The first quadrant, where both \( x \) and \( y \) are positive, is most relevant for the \( f(x) = \sqrt{x} \) function.
• It consists of a horizontal axis (x-axis) and a vertical axis (y-axis).
• Each point on the plane is represented by a pair \( (x, y) \) where \( x \) is the horizontal value and \( y \) is the vertical value.

Using the coordinate plane, you can graphically represent the relationship between \( x \) and \( y \).

Plotting Points

Plotting points accurately on the coordinate plane is fundamental to graphing functions. Here's how to correctly plot points:

• Start by identifying the coordinates. For example, the point \( (1, 1) \) has coordinates where \( x = 1 \) and \( y = 1 \).
• Move horizontally to the \( x \)-coordinate value on the x-axis.
• From that point, move vertically to the \( y \)-coordinate value.
• Mark the point where these values intersect.
• Repeat this for all key points. In this case, plot \( (0, 0) \), \( (1, 1) \), and \( (4, 2) \).
• Once you have all key points plotted, draw a smooth curve that passes through the points to represent the function.

These steps ensure that your graph is accurate and easy to interpret.