Question

Graph the function given in the following table. $$ \begin{array}{|c|r|r|r|r|r|r|r|r|r|} \hline \boldsymbol{x} & -7 & -6 & -3 & -1 & 0 & 2 & 5 & 6 & 8 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 4 & -1 & 0 & 7 & -2 & 6 & 2 & -4 & 1 \\\ \hline \end{array} $$

Short answer

Plot the points (-7, 4), (-6, -1), (-3, 0), (-1, 7), (0, -2), (2, 6), (5, 2), (6, -4), (8, 1) and connect them with a smooth curve.

Step-by-step solution

Identify Coordinates

Identify the coordinates from the table. Each column represents a point \(\boldsymbol{(x, f(x))}\). The points are: \((-7, 4)\), \((-6, -1)\), \((-3, 0)\), \((-1, 7)\), \((0, -2)\), \((2, 6)\), \((5, 2)\), \((6, -4)\), \((8, 1)\).

Set Up Axes

Draw a coordinate plane with a horizontal x-axis and a vertical y-axis. Ensure both axes include all values from the table.

Plot Each Point

Plot each point on the coordinate plane: \((-7, 4)\), \((-6, -1)\), \((-3, 0)\), \((-1, 7)\), \((0, -2)\), \((2, 6)\), \((5, 2)\), \((6, -4)\), \((8, 1)\). For each point, start at the x-value, move vertically to the corresponding y-value, and place a dot.

Connect the Dots

After all points are plotted, connect them with a smooth curve. This curve represents the function \(\boldsymbol{f(x)}\). Make sure to follow the pattern of the points.

Label the Graph

Clearly label the axes and points. Include arrows to indicate that the axes extend infinitely and label the function \(\boldsymbol{f(x)}\).

Key concepts

plotting coordinates

Plotting coordinates is a fundamental skill in graphing functions. It involves identifying and placing points on a coordinate plane. Each point \(x, y\) represents a specific pair of values from the given data set. For instance, the point \((-7, 4)\) means that when \(x = -7\), \(y = 4\).

Once you have a list of points, start plotting them on the graph. Begin with the x-coordinate: move horizontally from the origin. Then, move vertically to the y-coordinate and place a dot at the intersection.

When plotting multiple points, ensure accuracy by double-checking their positions. This will lead to a more accurate representation of the function when you connect the dots.

coordinate plane

A coordinate plane consists of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin \((0, 0)\). Each axis is marked with evenly spaced numbers, both positive and negative, to measure distance.

• The x-axis: It runs left to right (or right to left) and measures the horizontal distance of points.
• The y-axis: It runs up and down (or down and up) and measures the vertical distance of points.

When setting up a coordinate plane for specific data points, ensure that both axes cover all the values in your data set. Use appropriate scale to fit all points on the graph. It is also helpful to draw grid lines for better precision.

Lastly, include arrows at the ends of the axes to indicate that they extend infinitely. Clearly label each axis and mark intersecting lines to create an organized space for plotting.

smooth curve

Once all points are plotted accurately, the next step is to connect them with a smooth curve. This curve reflects the function's overall behavior.

The key to drawing a smooth curve is to interpret the pattern of the points. Do not simply connect the dots with straight lines. Instead, gently draw lines that flow through the points naturally.

Consider the function's possible trends like increasing, decreasing, or more complex behaviors. This helps ensure that the curve accurately represents the function's values.

A smooth curve gives a clear visual representation of the function, allowing you to observe crucial aspects like slope, intercepts, and continuity. Always take your time to ensure the curve captures the function’s nature accurately.

Don't forget to label the function and key points for clarity. This makes the graph easy to understand and interpret.