Problem 10 - College Algebra

Question

Graph the model. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 0.694 & 0.833 & 1 & 1.2 & 1.44 & 1.728 & 2.074 & 2.488 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Plot the coordinate points and connect them to form the graph of the data.

Step 1: Understand the Data

The table provides a list of x-values and their corresponding function values, \(f(x)\). These values will be used as coordinates \((x, f(x))\) in the graph.

Step 2: Identify the Coordinate Pairs

Extract the coordinate pairs from the table: \((-2, 0.694)\), \((-1, 0.833)\), \((0, 1)\), \((1, 1.2)\), \((2, 1.44)\), \((3, 1.728)\), \((4, 2.074)\), and \((5, 2.488)\).

Step 3: Set Up the Graph

Create a set of axes on graph paper or using graphing software. Label the x-axis for the range of \(x\) values (-2 to 5) and the y-axis for the range of \(f(x)\) values (0.694 to 2.488).

Step 4: Plot the Points

Carefully place a point on the graph for each coordinate pair \((x, f(x))\) identified. For example, the point \((-2, 0.694)\) will be to the left of the origin and slightly below the horizontal middle.

Step 5: Connect the Points

Draw a smooth curve or line through the points to visualize the function. Ensure that the curve reflects the trend shown by the points.

Key Concepts

Understanding Coordinate Pairs

Coordinate pairs are fundamental to graphing functions effectively. In any table of values like the one provided, each pair consists of an x-value and its corresponding f(x) value, often notated as \((x, f(x))\).
These pairs represent the exact location of a point on a graph. Think of a coordinate pair like an address in a city; it tells you exactly where to go.
To use these coordinate pairs, you'll start by identifying them in your data. Take a look at the table and list down each pair systematically.

• For instance, from the table, one of the coordinate pairs is \((-2, 0.694)\).
• Another is \((0, 1)\).

Listing them helps in organizing your graphing process and ensures precision when placing each point on your graph.

Graphing Software

Graphing software can greatly simplify the task of transforming coordinate pairs into a visual graph. Software like Desmos, GeoGebra, or more advanced options such as MATLAB allow you to input coordinate pairs and automatically generate accurate graphs.
These tools are beneficial, especially when dealing with complex functions or a large number of data points.

• They reduce human error when drawing graphs by hand.
• Offer flexibility in adjusting graph formats (like zooming in on specific areas).
• And often provide additional features such as plotting a line of best fit.

While graph paper remains a reliable classic, utilizing graphing software can save time and help visualize complex data more clearly.

Setup of Graph Axes

Proper setup of the graph axes is crucial. It's the framework of your graph and dictates how easy your data is to interpret. Start by drawing the x-axis and the y-axis perpendicularly.
To label them correctly, look at your coordinate pairs. For \(x\)-values between -2 and 5, ensure the x-axis accommodates this range. Similarly, for \(f(x)\)-values between 0.694 and 2.488, label your y-axis accordingly.

• Mark equal intervals on each axis, ensuring they are scaled so that each interval represents the same unit or value.
• Include labels and units if necessary.

Setting up axes this way ensures uniformity and clarity, making your graphs more intuitive to read.

Plotting Points on the Graph

Once you've set up your axes and identified your coordinate pairs, it's time to plot. This step is about accurately translating the pairs onto your graph. Begin with the first coordinate pair such as \((-2, 0.694)\).
Find \(-2\) on the x-axis and 0.694 on the y-axis and mark where these values intersect.

• Use a gentle pressure pencil for accuracy and adjustments.
• Repeat this for each coordinate pair.
• Take your time with each one to ensure you're placing them precisely.

After all points are plotted, connecting these with a smooth curve will help visualize the trend in your data. Proper plotting will make reading and interpreting the graph easier, providing insights into the function's behavior.